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††institutetext: Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, DK-2010, Copenhagen, Denmark

# Neutrinoless double-beta decay at colliders: interference between Majorana
states

Jonathan L. Schubert, Oleg Ruchayskiy<EMAIL_ADDRESS>
###### Abstract

Heavy neutral leptons (HNLs) are hypothetical particles able to explain
several puzzles of fundamental physics, first and foremost — neutrino
oscillations. Being _sterile_ with respect to Standard Model interactions,
these particles admit Majorana masses, allowing for violation of the total
lepton number. Lepton number violating (LNV) processes thus become a key
signature of HNLs, pursued by many experiments. In this work we demonstrate
that if HNLs are the sole origin of neutrino masses, destructive interference
between Majorana states suppresses the same-sign di-lepton signal. In the
phenomenologically interesting case of large HNL couplings, such a suppression
is akin to the cancellation of HNLs’ contributions to neutrino masses.
Nevertheless, the signal can be much larger than coming from the Weinberg
operator alone. We identify regions of the parameter space of such realistic
HNL models where the LNV signal is maximised at the LHC and future FCC-hh. Our
results are obtained within the effective W approximation which allows for
analytic treatment and gives a clear dependence on the model parameters.
Although approximate, they are argued to be correct within a factor of few.

##  1 Introduction

Searches for processes forbidden by the symmetries of the Standard Model
provide promising avenue of probing for new physics. Lepton Number Violating
processes (LNV) have always been among prime candidates for such searches (see
e.g.Goeppert-Mayer:1935uil ; Pontecorvo:1957vz ; Pontecorvo:1968wp ;
Weinberg:1979sa ; Wilczek:1979hc ; Lazarides:1980nt ; Mohapatra:1986bd ). The
motivation for it comes from the century-old observation of Ettore Majorana
Majorana:1937vz  that neutral particles admit a different type of mass,
_Majorana mass_. A main experimental approach to LNV searches is the
neutrinoless double-beta decay ($0\nu\beta\beta$, see DellOro:2016tmg ;
Cirigliano:2022oqy  and refs. therein). Other LNV probes include experiments
with muonic atoms SINDRUMII:1998mwd  and LNV decays of mesons and
$\tau$-leptons (see e.g. Belle:2012unr ; NA62:2019eax ; BaBar:2012eip ;
LHCb:2013hxr ). Collider searches, such as $pp\to\ell^{\pm}\ell^{\pm}jj$
ATLAS:2011izm ; ATLAS:2012ak ; CMS:2012wqj ; ATLAS:2015gtp ; CMS:2015qur ;
CMS:2016aro ; ATLAS:2018dcj ; CMS:2018agk ; CMS:2018jxx ; CMS:2018iye ;
LHCb:2020wxx , provide a powerful tool to probe for LNV, especially in the
sector involving particles other than electrons. The appeal of LNV searches
lies in their potential to deduce the presence of more massive particles than
can be accessed directly.

In this work we consider the _type-I seesaw model_ Minkowski:1977sc ;
Yanagida:1979as ; Glashow:1979nm ; Gell-Mann:1979vob ; Mohapatra:1979ia ;
Mohapatra:1980yp ; Schechter:1980gr ; Schechter:1981cv  in which the lepton
number violation comes from _heavy neutral leptons_ (HNLs). The parameters of
these HNLs are chosen so as to generate (Majorana) masses of neutrinos and
explain neutrino oscillations. The type-I seesaw model contains two sets of
Majorana particles: HNLs with masses $m_{\mathrm{N}_{I}}$ anywhere from
$\mathrm{e}\mathrm{V}$ to $10^{15}\,$\mathrm{G}\mathrm{e}\mathrm{V}$$; and
mass states of active neutrinos with $m_{\nu_{i}}$ being in sub-eV range as
dictated by neutrino oscillations Esteban:2020cvm  in combination with
cosmological measurements (see Dvorkin:2019jgs  and references therein). HNLs
interact with $W^{\pm}$, $Z^{0}$ and the Higgs boson similarly to the Standard
Model neutrinos but with the couplings suppressed by the flavour-dependent
mixing angles ${\Theta}_{\alpha I}$.111Our notations: flavour index:
$\alpha=\\{e,\mu,\tau\\}$; HNLs are enumerated with the index
$I=1,\dots,\mathcal{N}$. HNL masses: $m_{\mathrm{N}_{I}}$; the elements of the
$3\times\mathcal{N}$ matrix ${\Theta}_{\alpha I}$ describes interactions of
$I^{th}$ HNL with the flavour $\alpha$.

Many theoretical proposals on how to probe LNV signatures of HNLs have been
put forward (see e.g. Dicus:1991fk ; Datta:1993nm ; Ali:2001gsa ;
Panella:2001wq ; deGouvea:2007qla ; delAguila:2005pin ; delAguila:2006bda ;
Han:2006ip ; delAguila:2007ua ; Chen:2008qb ; FileviezPerez:2008wbg ;
Chao:2009ef ; Frandsen:2009fs ; Altarelli:2010gt ; Ibarra:2011xn ; Han:2012vk
; Angel:2012ug ; Helo:2013ika ; Helo:2013dla ; deGouvea:2013zba ; King:2014nza
; Deppisch:2015qwa ; Ng:2015hba ; Das:2016hof ; Anamiati:2016uxp ; Das:2017nvm
; Abada:2017jjx ; Cai:2017mow ; Chun:2019nwi ; Drewes:2019byd ; Tastet:2019nqj
; DeVries:2020jbs ; Gargalionis:2020xvt ; deGouvea:2021rpa ; Zhou:2021lnl ;
Aoki:2020til ; Fuks:2020att ; Fuks:2020zbm ; Zapata:2022qwo  for an incomplete
list of references). In this work we will concentrate on a single signature of
HNLs at hadron colliders: the process with two same-sign leptons and two jets
(and no missing energy) in the final state, $pp\to\ell^{\pm}\ell^{\pm}jj$. The
corresponding process may be mediated either via HNLs in the $s$-channel
(Drell-Yan process, figure 1b delAguila:2007qnc ; Degrande:2016aje ) or via
the direct analog of $0\nu\beta\beta$ with HNLs in $t$-channel — _$W$ boson
fusion_ (WBF), figure 1a, Datta:1993nm ; Ali:2001gsa ; Panella:2001wq ;
Chen:2008qb ; Fuks:2020att . Most of the previous LHC searches ATLAS:2011izm ;
ATLAS:2012ak ; CMS:2012wqj ; ATLAS:2015gtp ; CMS:2015qur ; CMS:2016aro ;
ATLAS:2018dcj ; CMS:2018agk ; CMS:2018jxx ; CMS:2018iye ; LHCb:2020wxx
concentrated on the process shown in figure 1b. The contribution of a single
HNL to the WBF process and the corresponding Monte Carlo level analysis of the
LHC sensitivity to such a signal were considered recently in Fuks:2020att . It
was found that the LHC sensitivity to these types of searches may be more
prominent than the Drell-Yan process for HNLs with masses
$m_{\mathrm{N}}\gtrsim$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. The
corresponding analysis was performed recently by the CMS collaboration
CMS:2022rqc .

However, in realistic HNL models, responsible for neutrino masses, the
situation is more complicated. Indeed, experimentally two mass splittings in
the light neutrino mass states have been measured Esteban:2020cvm . Therefore,
_at least two_ HNLs should be considered in the framework of the type-I seesaw
model to explain neutrino data. This means that even the simplest realistic
model contains _four_ Majorana particles: two light (active) neutrinos and two
HNLs.222To avoid potential confusion, we remind that in the type-I seesaw
model, two non-degenerate Majorana states are sufficient to account for two
light neutrino mass states. In _Inverse_ Mohapatra:1986aw ; Mohapatra:1986bd ;
Bernabeu:1987gr  or _Linear_ Akhmedov:1995ip ; Akhmedov:1995vm  Seesaw models
at least four HNLs with opposite CP phases are required. All these four
Majorana particles contribute coherently to the same LNV process and their
interference may be important.

An example of such an interplay is known from the studies of the similar
process – neutrino-less double beta decay ($0\nu\beta\beta$)
Cirigliano:2022oqy ; DellOro:2016tmg . On the one hand, the contributions to
the process of light and heavy states generically interfere destructively such
that the resulting signal, expressed via _effective neutrino mass_
$m_{\beta\beta}$ is smaller than the one, provided by the light Majorana
neutrinos alone, $m_{\beta\beta}^{\nu}$.333The literature on the subject is
vast, see e.g. Halprin:1983ez ; Benes:2005hn ; Bezrukov:2005mx ;
Blennow:2010th ; Mitra:2011qr ; Lopez-Pavon:2012yda ; Asaka:2013jfa ;
Faessler:2014kka ; Lopez-Pavon:2015cga ; Hernandez:2016kel ; Drewes:2016lqo ;
Asaka:2020lsx ; Asaka:2020wfo ; Asaka:2021hkg . On the other hand,
enhancements of the signal as compared with $m_{\beta\beta}^{\nu}$ is possible
for some values of HNL parameters Blennow:2010th ; Ibarra:2010xw ;
Ibarra:2011xn ; Mitra:2011qr ; Lopez-Pavon:2012yda ; Lopez-Pavon:2015cga ;
Hernandez:2016kel ; Drewes:2016lqo , especially if the spectrum of HNLs is
hierarchical Asaka:2020lsx ; Asaka:2020wfo ; Asaka:2021hkg .444This, however,
may require fine-tuning between tree level and one-loop contributions to
neutrino masses, see below. The question, therefore, arises — _how do light
and heavy Majorana degrees of freedom interplay in case of the WBF process at
colliders_? Does cancellation occur between the states? Can the signal get
enhanced for some values of HNL parameters? This paper is devoted to answering
these questions.

$u$$d$$u$$d$$u$$d$$\ell^{+}$$\ell^{+}$$u$$d$$u$$d$$u$$d$$W$$W$$\mathrm{n}$ (a)
t-channel diagram

$q$$\overline{q^{\prime}}$$\ell^{+}$$\ell^{+}$$q$$\overline{q^{\prime}}$$W$$\mathrm{n}$$W$
(b) s-channel diagram

Figure 1: Two kinds of processes leading to same-sign leptons plus jets
signatures in $pp$ collisions. Additional diagrams, originating from the
interchange of the final states are not shown. The symbol $\mathrm{n}$
represents both light and heavy Majorana states. Panel
LABEL:sub@fig:main_process: Majorana particles in $t$-channel ($W$ boson
fusion process). Panel LABEL:sub@fig:s-channel_lljj: Majorana particles in
$s$-channel (Drell-Yan process). The contribution from two processes are
incoherent because of the antiquarks in the final state of process
LABEL:sub@fig:s-channel_lljj as compared to LABEL:sub@fig:main_process.

#### Plan of the paper:

The paper is organised as follows. In section 2 we argue that in order to
estimate the contributions of Majorana particles to the
$pp\to\ell^{\pm}\ell^{\pm}jj$ cross section, it is sufficient to consider
same-sign $WW$ scattering to leptons. This makes our treatment analytic and
allows to explore the dependence of the cross section on the parameters of the
model. We then show that if HNLs are responsible for neutrino masses, the
contributions of several HNLs can cancel each other (Appendix B further
substantiates this statement). Based on the analysis of section 2 we can
anticipate in what parts of the parameter space the cross section can
nevertheless become sizeable.

In section 3 we compute same-sign $WW$ scattering in a model with two HNLs. We
show that, as expected, quasi-degenerate HNLs give negligible contribution to
LNV processes, while for hierarchical HNLs the contribution can become
sizeable. We substantiate the simplified treatment of section 3 by exact
computations in Appendix C.

Our results are summarised in section 4. They demonstrate that the total cross
section for the process $pp\to\ell^{\pm}\ell^{\pm}jj$ is at best
$\mathcal{O}($1\text{\,}\mathrm{f}\mathrm{b}$)$ at $\sqrt{s_{\text{\sc
lhc}}}\simeq$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, making its
exploration challenging even during the high-luminosity LHC (HL-LHC) phase.
Owing to the higher centre of mass energy — and resultantly larger
$pp\to\ell^{\pm}\ell^{\pm}jj$ cross section — we find that the proposed Future
Circular Collider, during its $\sqrt{s_{\text{\sc
fcc}}}\simeq$100\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ hadron-hadron phase
(FCC-hh), could produce competitive bounds in the
$m_{\mathrm{N}}\sim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$ regime.
Both these results require the HNL masses to differ by a factor of few.

We also discuss (section 5.1) whether, as commonly believed, LNV signals in
the type-I seesaw model are proportional to neutrino masses. Appendices A,
LABEL:, B, LABEL:, C, LABEL: and D contain additional computations as well as
basic definitions, needed for completeness of our results. In Appendix E we
present possible points of departure for future works.

##  2 Main idea and results

$W^{+}$$W^{+}$$\ell^{+}$$\ell^{+}$$k_{1}$$k_{2}$$\nu_{L}$$\nu_{L}$$p$$\nu_{R}$
Figure 2: Feynman Diagram of the two same sign $W$-bosons to two same sign
leptons in the flavour basis to first order in ${\Theta}^{2}$. The crosses
mark a transition from standard model neutrino to right handed sterile
neutrino, where the coupling is the Dirac mass $(m_{D})_{I\ell}$.

The process in figure 1a receives contributions from all Majorana particles
(HNLs and light neutrinos). An analysis of their relative contribution to the
LNV processes and potential interference can be performed at the level of a
$2\to 2$ process

$W^{\pm}W^{\pm}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}$ (2.1)

(where $\ell_{\alpha,\beta}^{\pm}$ are any of the Standard Model charged
leptons, not necessarily of the same flavour). Once the cross section for the
process in eq. 2.1 is known, one can approximately reconstruct the full
$pp\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}jj$ cross section by using the
_effective W-boson approximation_ (or EWA)555The PDFs of this approximation
are derived at leading QCD order, but includes further approximations (for
discussion see Appendix D). Monte Carlo studies Fuks:2020att  demonstrate that
the $k$-factor (NLO vs. LO correction) is $\lesssim 1.5$ for the processes in
question, supporting this approach. Dawson:1984gx ; Kane:1984bb ;
Kunszt:1987tk :

$\sigma(pp\to\ell^{\pm}\ell^{\pm}jj)=\\\
2\sum_{\begin{subarray}{c}\lambda,\lambda^{\prime}=\\{\mathrm{T,L}\\}\\\
\xi=\pm\
\end{subarray}}\int\differential{x_{1}}\differential{x_{2}}f_{W^{\xi}}^{\lambda}(x_{1})f_{W^{\xi}}^{\lambda^{\prime}}(x_{2})\sigma^{\lambda\lambda^{\prime}}(W^{\xi}W^{\xi}\to\ell^{\xi}\ell^{\xi})(x_{1}x_{2}s_{pp}).$
(2.2)

where ${x}_{1},{x}_{2}$ are the $W$’s Björken $x$, $s_{pp}$ is the $pp$ centre
of mass energy, and the sum goes over both signs of $W$-bosons ($\xi$) as well
as the transversal (T) and longitudinal (L) polarizations.

In this approximation, $W$-bosons are treated as polarised partons, and the
$pp$ level cross section is derived by folding the polarised cross sections of
the $WW$ scattering process,
$\sigma^{\lambda\lambda^{\prime}}(W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm})$,
with the corresponding parton distribution functions (PDFs)
$f_{W^{\pm}}^{\lambda}(x)$. More details are provided in the Appendix D. The
caveats and limitations of this approximation have been recently analysed in
Ruiz:2021tdt  (see also Kunszt:1987tk ; Ruiz:2015gsa ).

Going fast-forward, the results of our analysis are shown in figure 3 in terms
of parameter exclusion on the HNL mixing angle ${\Theta}_{\ell 1}$ at
$95\text{\,}\%$ c.l. without detector cuts and assuming background-free
searches for the high-luminosity LHC phase and FCC-hh. The details and
assumptions that went into these exclusion limits are presented in sections 3,
LABEL: and 4 below. For a qulaititative discussion of these results see
section 5.

Figure 3: Exclusion limits on the HNL mixing angle for WBF-mediated process
$pp\to\mu^{\pm}\mu^{\pm}+jj$ at the HL-LHC and FCC-hh (purple and orange
lines). The results are presented for two HNLs whose mass ratio equals to
$r_{\mathrm{N}}$, with ${\Theta}_{\ell 1}$ being the mixing angle of the
lighter HNL (see in-plot legends). The long-short dashed line indicates the
rescaling of the recent CMS dilepton search CMS:2022rqc  to HL-LHC phase (red-
grey). Short dash-dotted “$\mathrm{CCDY}+W\gamma$” lines are extrapolations of
the CMS tri-lepton searches CMS:2018CCDY  based on Alva:2014gxa  (light blue)
and a MG5_aMC simulation based on the ATLAS detector Pascoli:2018heg  (yellow-
grey). All limits are at $95\text{\,}\%$ c.l. Grey ends of the orange and
purple curves correspond to Yukawa couplings approaching non-perturbative
values $1<|F_{\ell 1}|<4\pi$. See text for details.

The qualitative analysis of the matrix element for eq. 2.1 is most readily
performed in the _flavour basis_. In this case only the right-chiral neutrinos
$\nu_{R}$ contain a Majorana mass term.666Section 3 below substantiates these
arguments by proper computation in the mass basis. Therefore, analysis of the
diagram in figure 2 includes all contributions and interference terms.

The amplitude of the $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ subprocess shown
in figure 2 is written as

$\mathcal{M}_{\text{ww}}^{\mu\nu}=-i\frac{g^{2}}{2}\bar{u}(k_{2})\gamma^{\mu}P_{R}\frac{1}{\not{p}}\sum_{I}\left[(m_{D})_{\alpha
I}\frac{\not{p}+m_{\mathrm{N}_{I}}}{p^{2}-m_{\mathrm{N}_{I}}^{2}}(m_{D})_{\beta
I}\right]\frac{1}{\not{p}}\gamma^{\nu}P_{L}v(k_{1})+(k_{1}\leftrightarrow
k_{2}).$ (2.3)

here $g$ is the weak coupling constant, $P_{R,L}$ are chiral projectors, $u,v$
are spinors of the outgoing leptons. Dirac ($m_{D}$) and Majorana
($m_{\mathrm{N}_{I}}$) masses are defined in Appendix A. We have implicitly
chosen a counterclockwise evaluation direction, the effects of which —
according to the Feynman rules for fermion number violating interactions
Denner:1992vza  — are twofold:

  * –

The spinor associated with the fermion going against the chosen direction is
charge conjugated.

  * –

The chiral projector in the vertex connecting to this fermion is transformed
according to $\Gamma^{\prime}=C\Gamma^{T}C^{-1}$, where $C$ is the charge-
conjugation matrix.

These rules result in the cancellation of the $\not{p}$ term in the numerator
of eq. 2.3, so that the amplitude is proportional to $m_{\mathrm{N}_{I}}$.
This cancellation can be anticipated from the fact that LNV effects are
enabled solely by the existence of the Majorana mass. We can rewrite eq. 2.3
in the compact form

$\mathcal{M}_{\text{ww}}^{\mu\nu}=-i\frac{g^{2}}{2}\biggl{(}\bar{u}(k_{1})\gamma^{\mu}\gamma^{\nu}P_{L}v(k_{2})\biggr{)}\times\sum_{I}{\Theta}_{\alpha
I}{\Theta}_{\beta
I}\left[\frac{m_{\mathrm{N}_{I}}^{3}}{t(t-m_{\mathrm{N}_{I}}^{2})}-(t\leftrightarrow
u)\right],$ (2.4)

where $t$ and $u$ are Mandelstam variables.

At this stage we can already anticipate an answer to the questions raised in
section 1: In the limit $|t|,|u|\ll m_{\mathrm{N}_{I}}^{2}$, the sum in eq.
2.4 reduces to the seesaw expression (see Appendix A):

$m_{\alpha\beta}^{(\nu)}\equiv\sum_{i=1}^{3}V_{\alpha i}m_{i}V_{\beta
i}=-\sum_{I}{\Theta}_{\alpha I}{\Theta}_{\beta I}m_{\mathrm{N}_{I}}.$ (2.5)

where $V_{\alpha i}$ are PMNS matrix elements and $m_{i}$ are light neutrino
masses. This means that for very heavy HNLs the total cross section will be
proportional to the neutrino masses, i.e. will be unobservable.

Similarly, for _almost degenerate_ HNLs ($m_{\mathrm{N}_{1}}\simeq
m_{\mathrm{N}_{2}}=m_{\mathrm{N}}$) eq. 2.4 becomes proportional to a linear
combination of $m_{\mathrm{N}}\sum_{I}{\Theta}_{\alpha I}{\Theta}_{\beta I}$
which again reduces to the light neutrino mass matrix (2.5). This result is
not surprising, as for two quasi-degenerate HNLs the total lepton number
becomes conserved Shaposhnikov:2006nn ; Kersten:2007vk .

However, in the case where $|t|,|u|\sim m_{\mathrm{N}_{I}}^{2}$ and/or HNLs
are non-degenerate, eq. 2.4 does not readily reduce to the seesaw relation and
thus is not proportional to the neutrino masses. Hence, _one can expect that
the largest LNV effect can be achieved for the HNLs with hierarchical masses._
The following sections substantiate this claim.

##  3 $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ scattering in the model with two
HNLs

HNLs with mass $m_{\mathrm{N}_{I}}$ and coupling ${\Theta}_{\ell I}$
contribute to neutrino mass matrix at the level ${\Theta}_{\ell
I}^{2}m_{\mathrm{N}_{I}}$. Therefore, to be consistent with neutrino data, one
has two distinct possibilities: either mixings ${\Theta}_{\ell I}$ are close
to the _seesaw line_ (see Appendix A):

$|{\Theta}_{\ell I}|^{2}\sim 5\times
10^{-14}\left(\frac{$\mathrm{T}\mathrm{e}\mathrm{V}$}{m_{\mathrm{N}}}\right)\;,$
(3.1)

or several HNLs cancel each other’s contributions to neutrino masses
Shaposhnikov:2006nn ; Kersten:2007vk . This latter case is phenomenologically
interesting, as HNL mixings can exceed the naive limit (3.1) by orders of
magnitude, making them accessible for direct searches. We will, therefore,
concentrate on this case.

###  3.1 Quasi-Dirac-like model – two HNLs with cancelling contribution to
neutrino mass

To simplify our treatment, we arrange the HNL mixing angles in such a way that
the individual contributions to the light neutrino masses cancel:777In this
Section we ignore the flavour structure and consider a generic flavour $\ell$.

$0=m_{\ell\ell}^{\nu}=\sum_{I=1}^{2}{\Theta}_{\ell I}^{2}m_{\mathrm{N}_{I}}.$
(3.2)

This allows getting rid of light neutrino contributions and analysing only
interference between HNL states. We will see below that the contribution to
the WBF process will be non-zero in this case, because for large HNL mixings,
neutrino contributions are subdominant. The effect from light neutrino masses
is presented in Appendix C. Finally, this parameter choice also allows us to
demonstrate that LNV can be prominent, even if neutrino masses are zero.

In the context of 2 HNLs, eq. 3.2 can be satisfied by the prescription of

${\Theta}_{\ell 2}=\pm i{\Theta}_{\ell
1}\sqrt{\frac{m_{\mathrm{N}_{1}}}{m_{\mathrm{N}_{2}}}}=\pm
i\frac{{\Theta}_{\ell 1}}{\sqrt{r_{\mathrm{N}}}},$ (3.3)

where we have introduced the _mass ratio_

$r_{\mathrm{N}}\equiv\tfrac{m_{\mathrm{N}2}}{m_{\mathrm{N}1}}\geq 1.$ (3.4)

We will refer to the HNL pair obeying LABEL:Eq.Masslessness_Condition with
$r_{\mathrm{N}}>1$ as the _quasi-Dirac-like_ (qDl) model.

In the limit of $r_{\mathrm{N}}\to 1$, known as the _Dirac-limit_ , the two
HNLs will form a single Dirac fermion. As a consequence, SM lepton number is
conserved and the LNV effects will cancel entirely. This is not the case if
$r_{\mathrm{N}}>1$ where, as we will see, LNV effects are non-zero. The qDl
model produces realistic predictions in the large mixing angle regime, which
we will present in the following. By comparison, the effect of light neutrino
masses is marginal (presented in Appendix C).

###  3.2 Cross section of the $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ process

The $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ cross-section depends on the $WW$
centre of mass energy $\sqrt{s_{\text{ww}}}$. This variable corresponds to
$x_{1}x_{2}s$ in the EWA approximation formula (2.2). In the following we set
$\sqrt{s_{\text{ww}}}\simeq$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ which
represents a typical value for the invariant mass of the $WW$ system in $pp$
scattering at the $13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ LHC (c.f.
Fuks:2020att ).

Averaging over $W$ polarisations contracted with the amplitude in eq. 2.3 we
can find the differential cross section of the
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ process which is shown for different
mass ratios (viridis colour scheme) in LABEL:Fig.2HNL_diffXSect_Massless. As
expected, the differential cross section decreases as $r_{\mathrm{N}}\to 1$
approximately as
${\differential{\sigma}}/{\differential{t}}\propto{(r_{\mathrm{N}}-1)^{2}}$.
The process fully vanishes in the case of exact HNL pair degeneracy (quasi-
Dirac limit). For $r_{\mathrm{N}}\gg 1$, the differential cross section in the
two HNL model approaches that of a single isolated HNL $\mathrm{N}_{1}$
(dashed orange), so that from $r_{\mathrm{N}}=10$ the two become
indistinguishable at the scales shown here. This is due to the fact that for
large HNL masses ($m_{\mathrm{N}_{2}}^{2}\gtrsim|t|_{\max}$) the denominator
of the heavier HNL propagator is always dominated by the mass term,
suppressing all $\mathrm{N}_{2}$ contributions by at least
$r_{\mathrm{N}}^{-1}$. For all $r_{\mathrm{N}}$, the numerical value of the
cross section is of the same order of magnitude for all $t$. This corresponds
to an evenly distributed angular dependence of the ejectiles. However, we do
observe that for a qDl pair closer to degeneracy, there is a slight tendency
toward back-to-back scattering. The opposite is true for a highly hierarchical
configuration.

Figure 4: Differential cross section of
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ scattering in a model with two HNLs
with cancelling contributions to the light neutrino masses (qDl). The lighter
HNL mass is kept at
$m_{\mathrm{N}_{1}}=$150\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, while the
heavier one is determined by the mass ratio $r_{\mathrm{N}}$, see eq. 3.4. The
mixing angle suppression $|{\Theta}_{\ell 1}|^{4}$ is factored out of the
result. The mass of the final leptons $\ell$ is neglected. The centre of mass
energy is $500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 5 shows the total cross section of
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ as a function of the lighter HNL mass
$m_{\mathrm{N}_{1}}$. For masses
$m_{\mathrm{N}_{1}}\lesssim\sqrt{s_{\text{ww}}}/10\simeq$50\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
we see a difference between hierarchical HNLs with $r_{\mathrm{N}}\sim 10$ and
the single HNL case. This is due to $m_{\mathrm{N}_{2}}$ being smaller than
$\sqrt{s_{\text{ww}}}$ in this regime, so that
$m_{\mathrm{N}_{2}}^{2}\lesssim|t|_{\max}$ and decoupling of a heavier HNL
does not happen. Hence, a cancellation between the two contributions can
occur.

Figure 5: Total cross section of $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$
scattering in a model with two HNLs with cancelling contributions to the light
neutrino masses (qDl). The cross section is divided by $|{\Theta}_{\ell
1}|^{4}$. The mass of the lighter HNL is shown as the x-axis, while the
heavier one is $r_{\mathrm{N}}$ times larger. The masses of the final leptons
$\ell$ are neglected.

The overall shape of the total cross section, as a function of mass
$m_{\mathrm{N}_{1}}$, exhibits maximal value for
$m_{\mathrm{N}_{1}}\sim\sqrt{s_{\text{ww}}}$. Trends for small and large
$m_{\mathrm{N}_{1}}$ can be directly understood from the amplitude in eq. 2.3
of the $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ process, which at small mass
scales as $m_{\mathrm{N}_{1}}$. At large masses it is suppressed as
$m_{\mathrm{N}_{1}}^{-1}$, which can also be observed in the cross section.

We note that the cross section in a realistic qDl model can be substantially
larger than that in a single HNL model with the mixing angles close to the
seesaw line (see Appendix B).

##  4 Results: “neutrinoless double-beta decay” at Colliders

Using the effective $W$ approximation, the results of the previous section are
translated into $pp$ level cross sections. Although our results were obtained
in the qDl model (i.e. in the limit where HNLs’ contribution to neutrino
masses cancels exactly even if $m_{\mathrm{N}_{1}}\neq m_{\mathrm{N}_{2}}$),
they are valid in realistic 2HNL models with non-zero neutrino masses. In
Appendix C, it is demonstrated that far from the seesaw line ($|{\Theta}_{\ell
I}|\gg|{\Theta}|_{\text{seesaw}}$, the light neutrino’s contributions to the
process become negligible compared to those of the HNLs. As we will see, only
HNLs with large mixing angles could lead to detection of WBF at the LHC or
FCC-hh, so that it is justified to consider the qDl model.888We also repeated
our analysis for a single HNL in a model ignorant of light neutrino masses. At
$WW$ level, our results match those of Fuks:2020att  while at $pp$ level they
differ by a factor of $\sim 1.6$ in the region of full validity according to
Appendix D. The origin of this factor could be related to known limitations of
EWAs, see Ruiz:2021tdt  for detailed discussion. This makes numerical
integration of the relevant matrix elements simpler and numerically stable.

###  4.1 Large Hadron Collider

Figure 6 shows the expected total cross section of $pp\to\ell^{+}\ell^{+}+jj$
for a centre of mass energy $\sqrt{s_{\text{\sc
lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. We remind that the
effective W approximation largely underestimates the cross-section for masses
_below few hundred GeV_ (see Appendix D below) and therefore our results only
apply for larger masses. Compared to figure 5 we see that the polarisation
decomposition and subsequent folding with the $W$ PDFs shifts the maximum from
around $m_{\mathrm{N}_{1}}\sim$0.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ to
$m_{\mathrm{N}_{1}}\sim$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. More
importantly, the $pp$ cross section is about 4 orders of magnitude smaller
than that of the $WW$. Beyond this, the general characteristics of the
$WW$-scattering case are directly translated into the $pp$ case.

Figure 6: Cross section of $pp\to\ell^{\pm}\ell^{\pm}jj$ as calculated in the
effective $W$ approximation in the model with two HNLs whose mass ratio is
equal to $r_{\mathrm{N}}$. The mass of the lighter HNL is shown as the x-axis,
while another one is $r_{\mathrm{N}}$ times heavier. Notations are the same as
in figure 5. The dependence on the mixing angle $|{\Theta}_{\ell 1}|^{4}$ is
factored out. Contribution of the light neutrinos is neglected (see section 3
for details). The centre of mass energy is assumed to be $\sqrt{s_{\text{\sc
lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$.

Using the $pp$ cross sections from LABEL:Fig.pp_2HNL_xSect_Massless, we can
estimate how many LNV events could be expected at the LHC. To this end, we
multiply the cross section by the maximal _admissible value_ of the mixing
angles (see Appendix A for details). Namely, _each mixing angle_
${\Theta}_{\ell 1}$ and ${\Theta}_{\ell 2}$ should obey the condition

$\bigl{|}{\Theta}_{\ell I}\bigr{|}\leq\left\\{\begin{aligned}
1&,&m_{\mathrm{N}_{I}}<\mathrm{vev}\\\
\frac{\mathrm{vev}}{m_{\mathrm{N}_{I}}}&,&m_{\mathrm{N}_{I}}>\mathrm{vev}\end{aligned}\right.$
(4.1)

for its respective mass. Notice that for $m_{\mathrm{N}_{2}}>\mathrm{vev}$ and
$r_{\mathrm{N}}>1$, the perturbativity condition of the HNL $\mathrm{N}_{2}$
provides the most stringent theoretical upper bound on $|{\Theta}_{\ell 1}|$
(under the assumption of eqs. 3.2 to 3.3).999This, in particular, demonstrates
that decoupling of one of the HNLs, while keeping neutrino masses small is not
possible – the condition for the theory to remain perturbative imposes
constraints on the mixings of non-decoupled lighter HNLs. This is an example
of the known violation of the “decoupling theorem” Appelquist:1974tg  in
type-I seesaw model, see e.g. recent discussion in Calderon:2022alb .

Figure 7 shows the expected maximal amount of events $N_{\max}$ at the HL-LHC
with an integrated luminosity of
$$3000\text{\,}{\mathrm{fb}}^{-1}$\equiv$3\text{\,}{\mathrm{ab}}^{-1}$$
HLLHCDesignReport2020  as a function of the lighter of two HNL masses
$m_{\mathrm{N}_{1}}$ for different mass ratios $r_{\mathrm{N}}$, as well as
for a single HNL case (dashed orange). The number of events reaches its
maximum at $m_{\mathrm{N}_{1}}\sim\mathrm{vev}/r_{\mathrm{N}}^{1/2}$. For
models with $m_{\mathrm{N}_{1}}<$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$N_{\max}$ is the largest for
$r_{\mathrm{N}}\gtrsim\mathcal{O}\left(10\right)$, while for
$m_{\mathrm{N}_{1}}>$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ $N_{\max}$
is largest for $r_{\mathrm{N}}\sim\mathrm{few}$. Notice that the differential
cross section for $r_{N}\geq 2$ looks similar to that of the single HNL.
Therefore, efficiencies of cuts will be similar to those estimated in
Fuks:2020att .101010For masses
$m_{\mathrm{N}_{1}}\sim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$ this
corresponds to a signal loss of around $40\text{\,}\%$.  We stress that for
all mass ratios $N_{\max}<1$ for
$m_{\mathrm{N}_{1}}\geq$1.2\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and that
the region above $1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ can only be probed
for $2\leq r_{\mathrm{N}}\lesssim 10$.

Figure 7: The maximal number of WBF events as calculated in the effective $W$
approximation in the model with two HNLs with mass ratio $r_{\mathrm{N}}$. The
mixing angle $|{\Theta}_{\ell I}|^{4}$ is maximised, according to eq. 4.1.
Notice that for
$m_{\mathrm{N}_{2}}=r_{\mathrm{N}}m_{\mathrm{N}_{1}}>\mathrm{vev}$, it is the
condition on the mixing angle $|{\Theta}_{\ell 2}|$ that dominates. The
overall efficiency of the signal is assumed $100\%$. The centre of mass energy
is $\sqrt{s_{\text{\sc lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and
the luminosity $3\text{\,}{\mathrm{ab}}^{-1}$.

Furthermore, we can estimate $95\%$ CL exclusion in the
$\left({\Theta}_{\ell\mathrm{N}},r_{\mathrm{N}},m_{\mathrm{N}}\right)$
parameter space. Below we report our results for $r_{\mathrm{N}}=2$ and
$r_{\mathrm{N}}=10$, as they represent the most promising mass ratio regime
according to LABEL:Fig.pp_2HNL_NMax_Massless. Bounds for $r_{\mathrm{N}}<2$
quickly deteriorate
$|{\Theta}_{\ell\mathrm{N}}|^{2}_{95\%}\propto|r_{\mathrm{N}}-1|^{-1}$. For
$r_{\mathrm{N}}>10$ growing fraction of the parameter space becomes excluded
due to perturbativity constraints, Section A.1. In a model with
$r_{\mathrm{N}}\sim\mathrm{few}$ (LABEL:Fig.pp_2HNL_U2_Massless_2), and
$100\%$ efficiency ($\epsilon\sim 1$, yellow), we find that a WBF search at
the HL-LHC can probe HNL masses $m_{\mathrm{N}_{1}}$ up to
$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$. This mass range is limited to
$m_{\mathrm{N}_{1}}\lesssim$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ for a
lower efficiency ($\epsilon\sim 0.1$, dashed grey). In a model with
$r_{\mathrm{N}}\sim\mathcal{O}\left(10\right)$
(LABEL:Fig.pp_2HNL_U2_Massless_10) the cross section per $|{\Theta}_{\ell
1}|^{4}$ is higher, but the range of accessible masses is smaller, due to the
perturbativity condition of eq. 4.1. As a result the maximally probed HNL mass
is $m_{\mathrm{N}_{1}}\lesssim$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
This limit drops to around $300\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ for a
lower efficiency.

(a)  (b)

Figure 8: _Optimistic_ exclusion limits on the $\mathrm{N}_{1}$ parameters via
WBF-mediated $pp\to\ell^{\pm}\ell^{\pm}jj$ events in the model with two HNLs
with mass ratio $r_{\mathrm{N}}$. The dashed red line indicates the
perturbativity limit of the Yukawa coupling $|F_{\ell 2}|=1$, while the solid
red line indicates $|F_{\ell 2}|=4\pi$. The exclusions limits are
$95\text{\,}\%$ CL assuming zero background. The centre of mass energy is
$\sqrt{s_{\text{\sc lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and
the luminosity $3\text{\,}{\mathrm{ab}}^{-1}$.

With efficiency dropping to $\epsilon\sim\mathcal{O}\left(10^{-2}\right)$, a
signal cannot be expected for any meaningful
$\left({\Theta}_{\ell\mathrm{N}},r_{\mathrm{N}},m_{\mathrm{N}}\right)$
combination.

###  4.2 Future Circular Collider

Sensitivity of the WBF process is maximal for $\sqrt{s_{\text{ww}}}\sim
m_{\mathrm{N}_{1}}$. The former increases with the $\sqrt{s}$ of $pp$
collision. Therefore the process under consideration can be expected to be
more efficient at the Future Circular Collider in the pp mode (FCC-hh)
FCC:2018byv . Using its projected centre of mass energy $\sqrt{s_{\text{\sc
fcc}}}=$100\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and machinery, developed
in this work, we can estimate the cross section of the
$pp\to\ell^{\pm}\ell^{\pm}jj$ process, shown in
LABEL:Fig.pp_2HNL_xSectFCC_Massless.

Figure 9: Cross section of $pp\to\ell^{\pm}\ell^{\pm}jj$ as calculated in the
effective $W$ approximation in the model with two HNLs whose mass ratio is
equal to $r_{\mathrm{N}}$. The mass of the lighter HNL is shown as the x-axis,
while another one is $r_{\mathrm{N}}$ times heavier. Notations are the same as
in figure 5. The dependence on the mixing angle $|{\Theta}_{\ell 1}|^{4}$ is
factored out. Contribution of the light neutrinos is neglected (see section 3
for details). The centre of mass energy is assumed to be $\sqrt{s_{\text{\sc
fcc}}}=$100\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$.

We find that the cross section reaches its maximum at
$m_{\mathrm{N}_{1}}\simeq$23\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$
(compared to
$m_{\mathrm{N}_{1}}\simeq$11.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ at LHC
energies). It is, furthermore, $\sim 2.5$ orders of magnitude larger at FCC
energies and reaches $\sim$1\text{\,}\mathrm{pb}$$ for a hierarchical HNL
pair.

With a target integrated luminosity
$30\text{\,}{\mathrm{ab}}^{-1}$Aleksa:2019pvl , this corresponds to a maximal
theoretically admissible event number $N_{\max}$ as shown in figure 10. Again,
we see that for $m_{\mathrm{N}_{1}}>r_{\mathrm{N}}^{-1/2}\mathrm{vev}$ Yukawa
perturbativity becomes the main theoretical constraint on the mixing angle
${\Theta}_{\ell 1}$, resulting in $r_{\mathrm{N}}\sim\mathrm{few}$ to yield
the largest $N_{\max}$.

Figure 10: The maximal number of WBF events as calculated in the effective $W$
approximation in the model with two HNLs with mass ratio $r_{\mathrm{N}}$. The
mixing angle $|{\Theta}_{\ell 1}|^{4}$ is maximised, according to eq. 4.1.
Efficiency of the detector is assumed $100\%$. The centre of mass energy is
$\sqrt{s_{\text{\sc fcc}}}=$100\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and
the luminosity $30\text{\,}{\mathrm{ab}}^{-1}$.

Due to the significantly larger cross section and higher integrated
luminosity, the relevant
$\left({\Theta}_{\ell\mathrm{N}},r_{\mathrm{N}},m_{\mathrm{N}}\right)$
parameter space covered by $pp\to\ell^{\pm}\ell^{\pm}jj$ at the FCC-hh opens
up. For both $r_{\mathrm{N}}=2$ and $r_{\mathrm{N}}=10$ (see
LABEL:Fig._pp_2HNL_U2FCC_Massless_2, LABEL: and
LABEL:Fig._pp_2HNL_U2FCC_Massless_10 respectively), physically relevant mixing
angles ${\Theta}_{\ell\mathrm{N}}$ are within reach for a qDl HNL pair with
$m_{\mathrm{N}_{1}}\sim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$
assuming a detection efficiency $\epsilon\geq 0.1$.

(a)  (b)

Figure 11: _Optimistic_ exclusion limits on the $\mathrm{N}_{1}$ parameters
via WBF-mediated $pp\to\ell^{\pm}\ell^{\pm}jj$ events in the model with two
HNLs with mass ratio $r_{\mathrm{N}}$. The dashed red line indicates the
perturbativity limit of the Yukawa coupling $|F_{\ell 2}|=1$, while the solid
red line indicates $|F_{\ell 2}|=4\pi$. The limits are obtained for
$95\text{\,}\%$ CL exclusions assuming zero background. The assumed detector
efficiency is $\epsilon$, the centre of mass energy $\sqrt{s_{\text{\sc
fcc}}}=$100\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and the luminosity
$30\text{\,}{\mathrm{ab}}^{-1}$.

##  5 Discussion and conclusion

In this work we analysed the collider probe for Majorana particles with masses
ranging from $\sim$50\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ to
$\sim$5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. The process that we
considered is a direct analog of the $0\nu\beta\beta$ decay —
$pp\to\ell^{\pm}\ell^{\pm}jj$ (figure 1a). Compared to previous works on the
subject (c.f. Dicus:1991fk ; Ali:2001pk ; Panella:2001wq ; Ng:2015hba ;
Fuks:2020att ) we concentrated on models where HNLs are solely responsible for
generating neutrino masses. All Majorana particles — HNLs and active neutrinos
— contribute to the process in question and we analysed their interference. We
demonstrated that such an interference is necessarily _destructive_ as a
consequence of smallness of neutrino masses, as compared to other relevant
energy scales. There are two limiting regimes depending on whether HNLs’
mixing angles $|{\Theta}_{\ell\mathrm{N}}|$ are comparable or much larger than
the naive “seesaw limit” $|{\Theta}|_{\text{seesaw}}^{2}=\frac{\sqrt{|\Delta
m_{\text{\rm atm}}^{2}|}}{m_{\mathrm{N}}}$ (see text around eq. A.8)

  1. 1.

For $|{\Theta}_{\ell\mathrm{N}}|\sim|{\Theta}|_{\text{seesaw}}$ cancellation
between HNLs and neutrino states can occur and the resulting cross section is
proportional to $m_{\nu_{i}}^{2}$.

  2. 2.

For $|{\Theta}_{\ell\mathrm{N}}|\gg|{\Theta}|_{\text{seesaw}}$ (while still
keeping neutrino masses small, as experimentally observed) the situation is
quite different. The contribution of active neutrinos is negligible, but
cancellation between HNL states does occur (similarly to the way it happens in
the neutrino mass matrix, see e.g. Kersten:2007vk ). On the one hand, the
cross-section is always smaller than that of a single HNL with the same
mixing. On the other hand, the cross-section can get enhanced as compared to
the naive scaling of $\sigma\propto m_{\nu}^{2}$ than could have occurred from
the Weinberg operator per se (c.f. Fuks:2020zbm ). This enhancement is roughly
by a factor of
$|{\Theta}_{\ell\mathrm{N}}|^{4}/|{\Theta}|_{\text{seesaw}}^{4}$ and occurs
only if HNLs are sufficiently far from the quasi-Dirac limit (i.e.
$m_{\mathrm{N}_{2}}/m_{\mathrm{N}_{1}}=r_{\mathrm{N}}\gtrsim\text{few}$).

  3. 3.

Lastly, large mass splitting $r_{\mathrm{N}}\gg 1$ does not allow to recover
the limit of a single HNL (as considered, e.g. in Fuks:2020att ). Naively, in
this case the heavier HNL $\mathrm{N}_{2}$ and its contribution to
$pp\to\ell^{\pm}\ell^{\pm}jj$ should disappear. This is not the case in our
realistic (and UV-complete) model (type-I seesaw with 2 HNLs). Indeed,
perturbativity demands that all Yukawa couplings are smaller than $\sim 1$.
For HNLs heavier than the Higgs’s VEV this condition together with the
requirement of the smallness of neutrino masses caps not only the value of the
mixing angle $|{\Theta}_{\ell 2}|$ but by extension also $|{\Theta}_{\ell
1}|$.

As a result, the $pp\to\ell^{\pm}\ell^{\pm}jj$ searches are most sensitive for
$r_{\mathrm{N}}\sim\text{few}$ at
$m_{\mathrm{N}_{1}}\sim$0.51\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$.111111Contribution
of the hierarchical HNLs to the $0\nu\beta\beta$ process have been studied
recently in Asaka:2021hkg ; Asaka:2020lsx ; Asaka:2020wfo . Even in this case
the cross section for the process $pp\to\ell^{\pm}\ell^{\pm}jj$ is about 1
order of magnitude smaller than that of a single HNL with the same mixing
angle. At the LHC, the best exclusion limit obtainable in this case would
reach $|{\Theta}_{\ell 1}|^{2}\sim$0.020.05\text{\,}$$ (assuming background-
free search, $100\%$ efficiency of the signal detection and the luminosity of
$$3\text{\,}{\mathrm{ab}}^{-1}$)$. In particular, even at the end of the high-
luminosity LHC phase ($3\text{\,}{\mathrm{ab}}^{-1}$ integrated luminosity)
these limits are non-competitive with those coming from the _electroweak
precision tests_ (EWPT) Fernandez-Martinez:2016lgt  or from non-observations
of lepton _flavour violating_ process, involving charged leptons
Calderon:2022alb . At the FCC-hh, the exclusion limit could be as low as
$|{\Theta}_{\ell 1}|^{2}\sim($25\text{\,}$)\times 10^{-4}$ for
$r_{\mathrm{N}}\sim\mathrm{few}$ in a mass range of
$m_{\mathrm{N}_{1}}\sim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$. These
bounds are competitive with EWPT exclusion. Furthermore, in the regime of
$m_{\mathrm{N}_{1}}\sim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$, they
could improve on exclusion bounds set by those of $\mathrm{CCDY}+W\gamma$
searches Pascoli:2018heg .

Throughout this paper we assumed background free searches. This is true only
approximately. Indeed, there are Standard Model processes that may lead, e.g.
to the appearance of two same-sign $W$’s in the final states, with subsequent
$W\to\ell\nu$ decays with low missing transverse momentum, see e.g.
Alboteanu:2008my ; Frandsen:2009fs . Additional sources of the background
include: mis-interpretation of jets as leptons (so-called _fake leptons_)
ATLAS:2010vza ; ATLAS:2014ffa ; Thusini:2017ztk ; processes with 3 or more
leptons, some of them escaping the detection; processes with the charge
misidentification, etc. At the same time, previous searches CMS:2012ggh ;
CMS:2022rqc  show that the backgrounds can indeed be made low by choosing
suitable discriminating variables. We leave the detailed studies to the future
work. However, we do not expect a drastic change of the results, as further
reduction of the signal due to cuts Fuks:2020att  will be in the same ballpark
as underestimation of the signal due to the EWA.

In summary: We explored the potential of the WBF signal at the LHC. We
concluded that for the planed HL-LHC upgrade it can only contribute meaningful
mixing bounds in a quasi-Dirac-like HNL model with HNL mass
$m_{\mathrm{N}_{1}}<$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ and the
corresponding mass $m_{\mathrm{N}_{2}}$ according to the mass ratio. For the
FCC-hh, the mass range can be extended to
$m_{\mathrm{N}_{1}}\sim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$
yielding competitive limits for mass ratios of
$r_{\mathrm{N}}\sim\mathrm{few}$ (see figure 3).

The already existing bounds derived from the WBF process CMS:2022rqc  can be
reinterpreted as bounds on a two HNL model in the highly hierarchical limit
owing to the fact that differential cross section for $r_{\mathrm{N}}\gg 1$
looks similar to the single HNL case and therefore the signal acceptance stays
roughly the same. We stress again that this does not correspond to a
decoupling in the traditional sense, as the “decoupled” and “non-decoupled”
HNLs’ mixing angles are explicitly related by the requirement that neutrino
masses remain small as observed experimentally.

###  5.1 Relation between LNV effects and small neutrino masses

Let us comment on the suppression of LNV effects as a consequence of smallness
of neutrino masses, a subject of discussion in many works in the past
Shaposhnikov:2006nn ; deGouvea:2007qla ; Kersten:2007vk ; Moffat:2017feq ;
Tastet:2019nqj ; Drewes:2019byd . The relation between the two signals would,
essentially, mean that every LNV effective operator deGouvea:2007qla  is
dependent on the Weinberg operator Weinberg:1979sa . Our results demonstrate
that this is not the case, as even in the qDl model, where neutrino mass is
exactly zero, the WBF signal remains finite. We do see, however, that the
destructive nature of HNL interference is related to the cancellation of their
contributions to the neutrino masses. In the Dirac-limit ($r_{\mathrm{N}}\to
1$ in our terms), both the smallness of neutrino masses and LNV collider
effects would be proportional to the same small perturbation (e.g. $\Delta
M/M$, Shaposhnikov:2006nn ; Kersten:2007vk ). However, in the hierarchical
case ($r_{\mathrm{N}}>\mathrm{few}$), this does not apply and the two kinds of
LNV operators become independent.

The condition $r_{\mathrm{N}}>\mathrm{few}$ implies that our results are not
directly applicable to models like the $\nu$MSM (where HNLs are highly
degenerate Asaka:2005pn ; Asaka:2005an ; Shaposhnikov:2006nn , see
Boyarsky:2009ix  for review). Rather, these results are applicable for
leptogenesis models with 3 HNLs Akhmedov:1998qx ; Drewes:2017zyw ;
Abada:2018oly  where the ratio $r_{\mathrm{N}}\sim\mathrm{few}$ and large
mixing angles are consistent with successful leptogenesis. We leave the
generalisation of our results for the case of 3 HNLs to future works.

The hierarchical HNL spectrum has its own drawbacks. As pointed out in past
literature Pilaftsis:1991ug ; Korner:1992zk ; Kersten:2007vk ;
AristizabalSierra:2011mn ; Lopez-Pavon:2012yda ; Pascoli:2013fiz ; Lopez-
Pavon:2015cga ; Haba:2016lxc ; Drewes:2019byd , the light neutrino mass states
generated in models with a highly hierarchical qDl HNL pair can exhibit a
significant running of the light masses. This makes these kinds of models
theoretically less appealing. Owing to the fact that even hierarchical HNLs
are not expected to provide a sizeable signal at the HL-LHC, we leave the
detailed exploration of this question to the future.

###### Acknowledgements.

We are grateful to Richard Ruiz and Benjamin Fuks for providing fruitful
feedback on the initial version of the manuscript and to Mads T. Frandsen for
his valuable input on the JLS.’s Master’s thesis Jonathan_Schubert_MSc . We
would also like to thank Kevin Urquía Calderón, Inar Timiryasov, as well as
members of the “NBI HNL group” for many discussions during this work. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (GA
694896) and from the Carlsberg Foundation. O.R. would like to thank the
Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for support via the
Centro de Excelencia Severo Ochoa Program under Grant CEX2020- 001007-S,
during the Extended Workshop “Neutrino Theories”, where this work developed.

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##  Appendix A Type I seesaw

In this section we summarise the main formulas of the type-I seesaw model with
the goal to fix the notations. For references see, e.g. Alekhin:2015byh ;
Abdullahi:2022jlv  and refs. therein.

Before Electro-Weak Symmetry Breaking (EWSB) the type-I seesaw Lagrangian is

$\displaystyle\mathcal{L}_{\mathrm{seesaw}}$
$\displaystyle=\mathcal{L}_{\mathrm{SM}}+\frac{i}{2}\nu_{RI}^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\nu_{RI}-\left(F_{\alpha
I}\right)^{*}\left(L_{\alpha}\cdot\tilde{\phi}\right)^{\dagger}\nu_{\mathrm{\scriptscriptstyle
R}I}-\frac{M_{I}}{2}{\nu}_{RI}^{T}\nu_{RI}+h.c.$ (A.1)

Here $\nu_{RI}$ are the new right-chiral singlet states with
$I=1,\dots,\mathcal{N}$ with associated Majorana masses $M_{I}$, $L_{\alpha}$
is the SM left chiral SU(2) doublet $L_{\alpha}=\begin{pmatrix}\nu_{\alpha}\\\
l_{\alpha}\end{pmatrix}_{L}$, where $\alpha=e,\mu,\tau$ and
$\tilde{\phi}_{a}=\varepsilon_{ab}\phi_{b}^{*}$, where $\phi$ is the Higgs
doublet. After EWSB, this can be written as
$\phi=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v\end{pmatrix}$, where $v$ is the
vacuum expectation value (VEV). And so
$\left(L_{\alpha}\cdot\tilde{\phi}\right)=\frac{v}{\sqrt{2}}\nu_{L\alpha}$
after EWSB. The terms

$\displaystyle\mathcal{L}_{D}$ $\displaystyle=-\left(F_{\alpha
I}^{\nu}\right)^{*}\frac{v}{\sqrt{2}}\nu_{L\alpha}^{\dagger}\nu_{RI}+h.c.$
(A.2)

are equivalent to Dirac mass terms with $(m_{D})_{\alpha
I}=\frac{v}{\sqrt{2}}\left(F_{\alpha I}\right)^{*}$. Diagonalising the mass
term leads to two types of Majorana states: $\mathcal{N}$ heavy Majorana
states $\mathrm{N}_{I}$ with masses

$m_{\mathrm{N}_{I}}\simeq M_{I}$ (A.3)

and three light Majorana neutrinos whose masses are given by the _seesaw
formula_

$\Bigl{(}V^{\dagger}\mathrm{diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})V\Bigr{)}_{\alpha\beta}\simeq\sum_{I}{\Theta}_{\alpha
I}{\Theta}_{\beta I}m_{\mathrm{N}_{I}}.$ (A.4)

Here $V_{i\alpha}$ is the PMNS matrix, while ${\Theta}_{\alpha I}$ are the
active-sterile mixing angles121212To make the notation more readable, we will
sometimes write ${\Theta}_{\ell\mathrm{N}}$ instead of ${\Theta}_{\alpha I}$.

${\Theta}_{\alpha I}\equiv\frac{F_{\alpha I}v}{M_{I}}.$ (A.5)

The Majorana scale is not fixed and can range from
$\sim$\mathrm{e}\mathrm{V}$$ to
$10^{15}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.

###  A.1 Perturbativity limit

The seesaw relation in eq. A.4 is derived under the assumption

$\text{Type-I seesaw:}\quad|{\Theta}_{\alpha I}|<1.$ (A.6)

However, for large HNL masses, $m_{\mathrm{N}_{I}}\gtrsim v$, the requirement
of perturbativity of the model described by eq. A.1, $|F_{\alpha I}|<1$
becomes more restrictive than eq. A.6 (see also Chanowitz:1978mv ;
Durand:1989zs ; Fajfer:1998px ; Ipek:2018sai ; Calderon:2022alb  for the
discussion of perturbativity in the Type I seesaw model). Indeed, owing to eq.
A.5 we get

$\text{Perturbativity:}\quad|{\Theta}_{\alpha I}|<\frac{v}{M_{I}}.$ (A.7)

The conditions in eqs. A.6, LABEL: and A.7 were used in section 4 when
deriving maximal number of events produced at colliders figure 7.131313Notice
that in a model as described in section 3, this requirement will be imposed on
the heavier of the two HNLs and entail a limit on ${\Theta}_{\ell 1}$ through
eq. 3.2. Owing to the extra loop factor $\frac{1}{(4\pi)^{2}}$ the
perturbativity limit (A.7) can be pushed up by a factor of $4\pi$.

Finally, it should be noted that a type-I seesaw model produces light neutrino
masses of the order $m\sim|{\Theta}|^{2}m_{\mathrm{N}_{1}}$. This defines a
minimal mixing angle, admissible for a given HNL mass, commonly referred to as
the _seesaw line_

$|{\Theta}|_{\text{seesaw}}^{2}=\frac{\sqrt{|\Delta
m_{\mathrm{atm}}^{2}|}}{m_{\mathrm{N}}}.$ (A.8)

##  Appendix B Single HNL Case – Seesaw Line

In a toy model including only 1 HNL, the smallness of the light neutrino mass
states can only be introduced by a sufficiently small mixing
${\Theta}_{\alpha\mathrm{N}}$. This model can not account for neutrino
oscillations, since it can maximally generate a single potentially degenerate,
non-zero, light mass state level Drewes:2019mhg . For demonstration purposes,
we will only consider a single non-zero light state with a PMNS-like mixing of
$1$. We will further generously set the value of the light neutrino mass to
$1\text{\,}\mathrm{e}\mathrm{V}$, which roughly corresponds to the current
direct upper bound on neutrino masses KatrinBounds2022 . To this effect, we
consider a mixing
$|{\Theta}_{\ell\mathrm{N}}^{2}|=$1\text{\,}\mathrm{e}\mathrm{V}$/m_{\mathrm{N}}$,
which yields a constant light mass state for a given $m_{\mathrm{N}}$
according to eq. 2.5.

Averaging over $W$ polarisations contracted with the amplitude in
LABEL:eq:WWFlavour_Amplitude we find the differential cross section of the
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ process considering both heavy and
light states. This is shown for an HNL mass of
$150\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ as the blue graph in figure 12.
Similarly, we can derive the differential cross section if only light (shown
in dash-dotted green) or heavy (dashed orange) mass states contributed to the
process.
It’s important to note that the isolated light mass state case yields a larger
differential cross section than both the isolated heavy case and the combined
case. This is true over the entire $t$ range, while at the extremal values the
isolated $\mathrm{N}$-case drops slightly and the combined case approaches the
$\nu$-case. That is to say, the isolated heavy case results in a relatively
even angular distribution, while the light case strongly favours back-to-back
scattering. Due to the cancelling nature of the two contributions, which is
the strongest while both $u$ and $t$ are large when compared to
$m_{\mathrm{N}}^{2}$, the combined differential cross section has an even
stronger tendency toward back-to-back scattering.

Figure 12: Differential cross section of
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ scattering in the case of a single HNL
with mass $m_{\mathrm{N}}=$150\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ as a
function the Mandelstam-variable $t$ at a centre of mass energy of
$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. The mixing angle here is a
function of the HNL mass, such that
$|{\Theta}^{2}_{\ell\mathrm{N}}m_{\mathrm{N}}|=$1\text{\,}\mathrm{e}\mathrm{V}$\gtrsim
m_{\nu_{e}}$. For further information see text.

Figure 13: Cross section of $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ scattering
in the case of a single HNL as a function of the HNL mass at a centre of mass
energy of $500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. The mixing angle here
is a function of the HNL mass as well, such that
$|{\Theta}^{2}_{\ell\mathrm{N}}m_{\mathrm{N}}|=$1\text{\,}\mathrm{e}\mathrm{V}$\gtrsim
m_{\nu_{e}}$. For further information see text.

Integrating the differential cross section with respect to $t$ we find the
total cross section, shown in figure 13. As prescribed by the mixing angle,
the isolated light mass state case (again in dash-dotted green) results in a
constant cross section with respect to $m_{\mathrm{N}}$. As expected from the
differential cross section the light line represents an upper limit to the
numerical value of the combined cross section (solid blue). For small HNL
masses ($m_{\mathrm{N}}^{2}\lesssim|t|_{\min}$) both heavy and light mass
states will have a near identical differential cross section, leading to a
vanishing combined cross section. For large HNL masses
($m_{\mathrm{N}}^{2}\gtrsim|t|_{\max}$) the denominator of the HNL propagator
is always dominated by the mass term. This, combined with the mixing
prescription, results in the cross section of the heavy case dropping as
$m_{\mathrm{N}}^{-4}$, and the combined cross section asymptotically
approaching the light line.

We would like to draw attention to the fact that the numerical values of the
cross section resulting in this model are vanishingly small. Consequently,
even given the uniqueness of the signature, there would be no hope of
detection of a process involving $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ in
terms of a single HNL model.

##  Appendix C Realistic model with two HNLs plus light Majorana states

Given the importance of the light mass states in the single HNL case, it is
prudent to investigate their relevance when it comes to the two HNL case as
well. An easy way to incorporate them into the model is by using the Casas-
Ibarra parametrisation of the mixing angles

${\Theta}=iU\mathrm{diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})^{1/2}\Omega\mathrm{diag}(m_{\mathrm{N}_{1}},\dots,m_{\mathrm{N}_{\mathcal{N}}})^{-1/2},$
(C.1)

which, by design, generates the light mass states $m_{\nu_{i}}$ through eq.
2.5. Here $\Omega$ is an orthogonal $3\times\mathcal{N}$ matrix holding all
degrees of freedom of the model that aren’t fixed by the PMNS matrix $U$, and
the mass states $m_{\nu_{i}}$, and $m_{\mathrm{N}_{I}}$. To account for
neutrino oscillations with 2 HNLs, we require the lightest neutrino mass state
to be massless. The most general explicit form of $\Omega$ in case of normal
hierarchy (NH) is given by

$\Omega=\quad\begin{pmatrix}0&0\\\ \cos\omega&\sin\omega\\\
-\xi\sin\omega&\xi\cos\omega\end{pmatrix},$ (C.2)

with $\xi=\pm 1$, and $\omega\in\mathbb{C}$. The parity $\xi$ can be chosen by
simultaneous redefinition of the fields and $\omega$ so that we will choose
$\xi=+1$ without loss of generality. In the following we present the numerical
values for the case of normal hierarchy with $\ell=e$.141414The results will
be equivalent for other choices up to orders of few in PMNS matrix entry
ratios. The real part of the Casas-Ibarra parameter $\omega$ only regulates
which PMNS entry talks to which HNL, while $\imaginary(\omega)$ determines the
absolute scale of the mixing angle.

(a)  (b)

(c)

Figure 14: Differential cross section of
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ scattering using the Casas-Ibarra
parametrisation of the HNL mixing angles with enhancement parameter
$\imaginary(\omega)$. Here,
$r_{\mathrm{N}}=\tfrac{m_{\mathrm{N}2}}{m_{\mathrm{N}1}}$ is the mass ratio.
The lighter of the two HNL masses
$m_{\mathrm{N}_{1}}=$150\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and the
centre of mass energy is $490\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 14 shows the differential cross section in a model with two non-zero
light and two heavy mass states for $\omega=0$ (no enhancement), $\omega=5i$
(moderate enhancement), and $\omega=15i$ (strong enhancement) respectively.
Here, the two HNL curves are both given for a mass $m_{\mathrm{N}_{1}}$ but
represent the two possible mixing angles of the Cassas-Ibarra parametrisation
determined by $\real(\omega)$. We see that, in the case of no enhancement, the
situation is rather similar to the single HNL case (figure 12), where the
differential cross section is dominated by the light states, and we get a
strong back-to-back scattering. The strongly enhanced case is essentially the
same as the quasi-Dirac-like model of two HNLs (figure 4), but with a mixing
angle slightly smaller than $1$.151515Indeed, it is directly comparable with
qDl with a mixing angle of ${\Theta}_{\ell
1}^{\mathrm{\textsc{qDl}}}=|{\Theta}_{\ell 1}^{\mathrm{CI}}(\omega=15i)|<1$
(see also figure 15). For large $r_{\mathrm{N}}$, the moderately enhanced case
exhibits the same characteristics as the strongly enhanced case. However, for
smaller $r_{\mathrm{N}}$ the combined cross section becomes comparable to that
of the isolated light mass states, so that non-trivial cancellations occur.

Figure 15 compares the $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ cross section
as evaluated in a fully realistic 2 HNL model with the same in the qDl
approximation. We see that for a maximal theoretically admissible mixing angle
there is no observable difference between the two results at the scales
presented here.161616As described in section 4, this means that the mixing
angle fully satisfies either the Seesaw or Yukawa expansion limit.

qDl approx.

Full model

Figure 15: Comparison of $W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ cross
sections between qDl and fully realistic two HNL neutrino sector model. The
data is shown at a maximally admissible mixing angle ${\Theta}_{\ell 1}$
according to eq. 4.1.

This shows that for large ${\Theta}$, qDl becomes a very good approximation of
a fully realistic model. By virtue of this equivalence, for our purposes, the
numerical stability of the prior far outweighs the usefulness of considering
the full model. This is especially true since only large mixing angles have a
chance of being detected at the LHC and FCC (see section 4).

##  Appendix D Effective W approximation

In the deduction of the effective $W$ approximation’s parton distribution
functions (PDFs) according to Dawson:1984gx  (see figure 16 for PDFs at
relevant centre of mass energies), one formally assumes decoherence between
the polarisation states of individual $W$ “partons”. This means that if
contributions due to different polarisation states are on a similar scale —
and thus the decoherence assumption is no longer valid — the results at $pp$
level are potentially underestimated by a factor of
$\mathrm{few}^{2}$.171717This is due to the error occurring in both $p$
involved in the $pp\to\ell^{\pm}\ell^{\pm}jj$ process.

(a)  (b)

Figure 16: Parton distribution functions of the effective $W$ approximation as
prescribed in Dawson:1984gx  at LHC<EMAIL_ADDRESS>and expected
FCC centre of mass energy<EMAIL_ADDRESS>The underlying quark
parton distribution functions used were taken from Martin2009 .

(a) $\lambda_{1},\lambda_{2}=\mathrm{L,L};\,s_{\text{ww}}=16m_{W}^{2}$ (b)
$\lambda_{1},\lambda_{2}=\mathrm{T,T};\,s_{\text{ww}}=16m_{W}^{2}$

(c) $\lambda_{1},\lambda_{2}=\mathrm{L,L};\,s_{\text{ww}}=s_{\text{\sc
lhc}}/100$ (d)
$\lambda_{1},\lambda_{2}=\mathrm{T,T};\,s_{\text{ww}}=s_{\text{\sc lhc}}/100$

Figure 17: Polarised cross sections of
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$. The polarisatinos
$\lambda_{1},\,\lambda_{2}$ are both longitudinal ($\mathrm{L,L}$, left
panels) or transversal ($\mathrm{T,T}$, right panels), while the centre of
mass energy is $4m_{W}=$225\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ (top
panels) or $\sqrt{s_{\text{\sc
lhc}}}/10=$1.3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ (bottom panels). We
assume a qDl model with mass ratio $r_{\mathrm{N}}$.

To investigate the validity of the approximation we will thus compare the
doubly longitudinally polarised ($\mathrm{LL}$) $WW$ cross section to the
doubly transversally polarised ($\mathrm{TT}$) case. As the folding with PDFs
involves an integration over different energy regimes, this is important at
all relevant centre of mass energies determined by the PDFs in figure 16.
Figure 17 shows this for for a centre of mass energy on the lower end of the
PDF spectrum ($s_{\text{ww}}=16m_{W}^{2}$, figures 17a, LABEL: and 17b) and on
the upper end ($s_{\text{ww}}=s_{\text{\sc lhc}}/100$, figures 17c, LABEL: and
17d). We note that for low end energies, the $\mathrm{LL}$ cross section
becomes dominant for
$m_{\mathrm{N}_{1}}\gtrsim$100200\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
while at larger energies this is the case for
$m_{\mathrm{N}_{1}}\gtrsim$10100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. As
$pp$ level validity is determined by validity at all relevant subenergies,
only a region of general $\mathrm{LL}$ dominance fully meets the assumptions
of the effective $W$ approximation. We thus expect accurate results in a
parameter space of
$m_{\mathrm{N}_{1}}\gtrsim$400\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, while
we expect to underestimate the $pp$ level cross section for smaller
$m_{\mathrm{N}_{1}}$.

Figure 18: Single isolated HNL $pp$ level WBF cross section at
$\sqrt{s_{\text{\sc lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ as a
function of HNL mass $m_{\mathrm{N}}$. This is shown for different lower
bounds on the integration parameters of the effective $W$ approximation
determined by $M_{WW}$ according to eq. D.1.

It has been, furthermore, pointed out in past literature that in order to
achieve numerical stability, the lower limit of the integration Ruiz:2021tdt
has to be set above a certain threshold, rather than to account for minimal
production energy. We investigate this by varying the lower limit on

$x_{1}x_{2}=\frac{M_{WW}^{2}}{s_{pp}}.$ (D.1)

This is shown in figure 18, where we have plotted the cross section of a
single isolated HNL for different values of $M_{WW}^{2}$ at
$\sqrt{s_{\text{\sc lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. For
small HNL masses
$m_{\mathrm{N}}\lesssim$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, there is
a noticeable dependence on the lower cut-off. We note that the cross section
for $M_{WW}^{\min}=$320500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ is largest
in this regime, even though we removed part of the integration interval for
positive definite functions with respect to e.g
$M_{WW}^{\min}=$200\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. The lines of
$M_{WW}^{\min}=$320\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$M_{WW}^{\min}=$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ coincide for all
mass values $m_{\mathrm{N}_{1}}$. For
$m_{\mathrm{N}}\gtrsim\mathrm{several}\,$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
the numerical values are identical for all
$M_{WW}^{\min}<$750\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, while the curve
of $M_{WW}^{\min}=$750\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ lies slightly
below the others. The curve representing
$M_{WW}^{\min}=$1\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ lies below the
others by a factor of $\mathcal{O}($10\text{\,}\%$)$ for lower
$m_{\mathrm{N}}$ converging with the rest at
$m_{\mathrm{N}}\simeq\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$.

Overall the effect of the cut-off is only significant in the regime of
$m_{\mathrm{N}}\lesssim$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. As the
focus of this work lies on the regime of
$m_{\mathrm{N}}\simeq$\mathrm{several}\,100\mathrm{few}\,1000\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
the influence is thus marginal. Nevertheless, we set the lower limit
$M_{WW}^{\min}=4m_{W}\simeq$320\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ as a
benchmark for this work, which corresponds to twice the minimal production
energy and is already in a numerically stable regime according to figure 18.

Figure 19: WBF at $pp$ level cross section for a single isolated HNL
($W^{\pm}W^{\pm}$ combined). Comparison of MG5_aMC Montecarlo detector
simulation by Fuks et al. Fuks:2020att  (dashed orange) and effective $W$
approximation (purple), shown at LHC centre of mass energy $\sqrt{s_{\text{\sc
lhc}}}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$.

For the case of a single HNL in a phenomenological type-I seesaw, figure 19
compares the resulting cross section deduced using the EWA without any
detector-motivated cuts with that given in the NLO in QCD WBF study by Fuks et
al. Fuks:2020att . As expected from the above discussion, the results differ
significantly (by a factor of $\sim 10$) for
$m_{\mathrm{N}_{1}}\sim\mathrm{few}\,$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
For masses in the $\mathrm{T}\mathrm{e}\mathrm{V}$-range, however, the results
become comparable within order $\mathrm{few}$ reaching order
$\mathrm{several}\,$10\text{\,}\%$$ for
$m_{\mathrm{N}_{1}}\sim\mathrm{several}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$. As
this coincides with the significant mass range of our study, the $pp$ level
results presented here can be understood as estimates within $\mathcal{O}(1)$.
We highlight that this only affects the absolute scale of our $pp$ level
results, while relative differences between different mass ratios
$r_{\mathrm{N}}$ stem from analytic treatment at $WW$ level, and will thus
hold at $pp$ level as well.

##  Appendix E Potential avenues for future same sign di-lepton searches at
colliders

Even though WBF does not look like a promising channel for HNL discovery at
the LHC at tree level, it remains to be seen if loop-order corrections to the
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ system could potentially boost the
signal. Especially penguin corrections to the $W\mathrm{N}\ell$ vertices
(similar to Calderon:2022alb ) and corrections to the Majorana propagator
(similar to those responsible for running of light neutrino masses) could be
of interest here.

Another important aspect to investigate is the question of how a tendency
toward front-to-back scattering in the $WW$ system for smaller mass ratios
translates into angular distributions of the full $pp$ scattering. This will
need to be investigated in a full on $2\to 4$ simulation of the process as
this phenomenon only occurs for
$m_{\mathrm{N}_{1}}<\mathrm{few}\,$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
which is outside the full validity of the EWA, while all angular dependence is
strongly suppressed for HNLs with
$m_{\mathrm{N}_{1}}\gtrsim\mathrm{few}\,$\mathrm{T}\mathrm{e}\mathrm{V}$$. If
indeed front-to-back scattering translates into large lepton pseudorapidities
at $pp$ level, the drop off for smaller $r_{\mathrm{N}}$ with respect to
realistic detection prospects could be even more severe than anticipated.
However, due to the detection limits at the (HL-)LHC, this question is more
relevant for FCC energy level searches.

Another avenue of exploration could be to consider three or more HNLs with
similar masses and mixing angles to see if significant LNV effects could be
realised in a realistic neutrino sector model. As this greatly opens up the
available parameter space, _the single HNL case would not hold as an upper
limit_.