aainstitutetext: School of Physics, Sun Yat-Sen University, Guangzhou 510275,
Chinabbinstitutetext: School of Physics Sciences, University of Chinese
Academy of Sciences, Beijing 100049, Chinaccinstitutetext: Center for Future
High Energy Physics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, Chinaddinstitutetext: School of Physics, Southeast
University, Nanjing 211189, Chinaeeinstitutetext: Deutsches Elektronen-
Synchrotron DESY, 22607 Hamburg, Germanyffinstitutetext: School of Electronics
and Information Engineering, Guangdong Ocean University, Zhanjiang 524088,
China.

#  Improving heavy Dirac neutrino prospects at future hadron colliders using
machine learning

Jie Feng111Moved to Massachusetts Institute of Technology (MIT), Cambridge,
Massachusetts 02139, USA. b Mingqiu Li b,c Qi-Shu Yan a,f Yu-Pan Zeng a Hong-
Hao Zhang d Yongchao Zhang c,e Zhijie Zhao<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, by using the machine learning methods, we study the
sensitivities of heavy pseudo-Dirac neutrino $N$ in the inverse seesaw at the
high-energy hadron colliders. The production process for the signal is
$pp\to\ell N\to 3\ell+E_{T}^{\rm miss}$, while the dominant background is
$pp\to WZ\to 3\ell+E_{T}^{\rm miss}$. We use either the Multi-Layer Perceptron
or the Boosted Decision Tree with Gradient Boosting to analyse the kinematic
observables and optimize the discrimination of background and signal events.
It is found that the reconstructed $Z$ boson mass and heavy neutrino mass from
the charged leptons and missing transverse energy play crucial roles in
separating the signal from backgrounds. The prospects of heavy-light neutrino
mixing $|V_{\ell N}|^{2}$ (with $\ell=e,\,\mu$) are estimated by using machine
learning at the hadron colliders with $\sqrt{s}=14$ TeV, 27 TeV, and 100 TeV,
and it is found that $|V_{\ell N}|^{2}$ can be improved up to ${\cal
O}(10^{-6})$ for heavy neutrino mass $m_{N}=100$ GeV and ${\cal O}(10^{-4})$
for $m_{N}=1$ TeV.

##  1 Introduction

Seesaw mechanisms provide natural explanations of the tiny neutrino masses.
However, in the canonical type-I seesaw Minkowski:1977sc ; Mohapatra:1979ia ;
Yanagida:1979as ; GellMann:1980vs ; Glashow:1979nm , the Dirac Yukawa
couplings connecting the left- and right-handed neutrinos are generally very
small $Y_{D}\lesssim 10^{-6}$ if the heavy right-handed neutrinos $N$ are at
or below the TeV-scale. With some specific flavor structures such as the
discrete symmetries (see reviews in e.g. Refs. Altarelli:2010gt ;
Ishimori:2010au ; King:2013eh ; Petcov:2017ggy ; Xing:2020ijf ;
Feruglio:2019ybq ), the Yukawa couplings could be much larger, or even of
order one, while fitting the neutrino masses $m_{\nu}\sim 0.1$ eV Zyla:2020zbs
. Then the heavy neutrinos can mix with the active neutrinos with a sizable
value, and can be produced at the high-energy hadron Datta:1993nm ; Han:2006ip
; Bray:2007ru ; delAguila:2007qnc ; Kersten:2007vk ; delAguila:2008cj ;
Atre:2009rg ; Chen:2011hc ; BhupalDev:2012zg ; Das:2012ze ; Dev:2013wba ;
Das:2014jxa ; Deppisch:2015qwa ; Gago:2015vma ; Dev:2015kca ; Das:2015toa ;
Kang:2015uoc ; Degrande:2016aje ; Das:2016hof ; Mitra:2016kov ;
Anamiati:2016uxp ; Antusch:2016ejd ; Das:2017pvt ; Das:2017nvm ;
Accomando:2017qcs ; Das:2017gke ; Cai:2017mow ; Bhardwaj:2018lma ;
Helo:2018qej ; Das:2018hph ; Deppisch:2018eth ; Pascoli:2018rsg ;
Cottin:2018nms ; Abada:2018sfh ; Drewes:2019fou ; Drewes:2021nqr , lepton
Fargion:1995qb ; delAguila:2005ssc ; delAguila:2005pin ; Das:2012ze ;
Antusch:2015mia ; Banerjee:2015gca ; Asaka:2015oia ; Antusch:2015gjw ;
Antusch:2016ejd ; Antusch:2016vyf ; Cai:2017mow ; Biswal:2017nfl ;
Antusch:2017pkq ; Hernandez:2018cgc ; Deppisch:2018eth ; Chakraborty:2018khw ;
Han:2021pun ; Drewes:2021nqr , and lepton-hadron Mondal:2015zba ;
Antusch:2016ejd ; Lindner:2016lxq ; Mondal:2016kof ; Cai:2017mow  colliders.
As a result of the Majorana nature of the heavy neutrinos, the “smoking-gun”
signal of heavy neutrinos at hadron colliders would be same-sign dilepton plus
two jets $\ell^{\pm}\ell^{\pm}jj$, which is clearly a lepton number violating
(LNV) signature beyond the standard model (SM) Pilaftsis:1991ug ; Datta:1993nm
; Almeida:2000pz ; Han:2006ip ; Bray:2007ru ; Atre:2009rg .

In light of the small Yukawa couplings in the electroweak scale type-I seesaw,
we consider the TeV-scale inverse seesaw, in which the small neutrino masses
are governed by a small $\mu_{S}$ parameter in the heavy neutrino sector
Mohapatra:1986aw ; Mohapatra:1986bd . In the basic version of inverse seesaw,
the heavy neutrino states are mass degenerate and form pseudo-Dirac fermion
pairs, while the Majorana components are highly suppressed by the $\mu_{S}$
parameter. Different from the canonical type-I seesaw, the Yukawa couplings of
heavy neutrinos with the active neutrinos can be of order one, thus mixing
sizably with the active neutrinos and inducing very rich phenomenology. Recent
studies of $N$ at the hadron colliders can be found e.g. in Refs.
delAguila:2008cj ; Chen:2011hc ; Das:2012ze ; BhupalDev:2012zg ; Das:2014jxa ;
Dev:2015kca ; Deppisch:2015qwa ; Das:2015toa ; Antusch:2016ejd ; Das:2016hof ;
Anamiati:2016uxp ; Das:2017pvt ; Das:2018hph ; Bhardwaj:2018lma ;
Drewes:2019byd . For a relatively light $N$, say with mass $m_{N}\lesssim 1$
TeV, the dominate production channel at hadron colliders is through the
charged-current Drell-Yan (DY) process $pp\to W^{\ast}\to\ell N$, with the
subsequent decay $N\to\ell W$, assuming $N$ is heavier than the $W$ boson.
Then the leptonic decays of $W$ boson in the final state will result in the
trilepton signatures at the leading-order, i.e. three charged leptons plus
significant missing transverse energy (MET). As the production of $N$ at high-
energy colliders is induced by the heavy-light neutrino mixing, the
corresponding mixing angle $|V_{\ell N}|^{2}$ can be directly measured, or
constrained, in the trilepton signatures Das:2014jxa ; Das:2015toa ;
Das:2016hof ; Antusch:2016ejd ; Das:2017pvt ; Das:2018hph ; Bhardwaj:2018lma .
This is largely complementary to the limits on $|V_{\ell N}|^{2}$ from the
high-intensity beam-dump experiments, meson decay data, beta decays,
neutrinoless double-beta decays, reactor neutrino experiments, neutrino
oscillation data, electroweak precision data (EWPD), direct high-energy
collider searches in other channels e.g. the process
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$, and the cosmological and
astrophysical observations Atre:2009rg ; Deppisch:2015qwa ; deGouvea:2015euy ;
Chrzaszcz:2019inj ; Bolton:2019pcu ; Fuks:2020att . Furthermore, heavy
neutrinos at the hundreds of GeV scale can be used to explain the muon $g-2$
discrepancy Cirigliano:2021peb .

In this paper, we apply machine learning (ML) to the searches of heavy pseudo-
Dirac neutrinos in the inverse seesaw at the High-Luminosity LHC (HL-LHC) 14
TeV, High-Energy LHC (HE-LHC) 27 TeV and future 100 TeV colliders such as the
Future Circular Collider (FCC-hh) FCC-hh  and the Super Proton-Proton Collider
(SPPC) Tang:2015qga . In particular, we use either the Multi-Layer Perceptron
(MLP) Fukushima:2013b  or the Boosted Decision Tree with Gradient boosting
(BDTG) 10.2307/2699986  technique in the Toolkit for Multivariate Data
Analysis (TMVA) package 2007physics…3039H  to define the multi-variate
estimators to separate the signal from SM backgrounds. Combining some
kinematic distributions of the charged leptons, MET and extra jets (see Figs.
4 to 7), it is found that the discrimination power of these two methods are
close (see Fig. 11). It is promising that the usage of ML can improve
significantly the prospects of heavy-light neutrino mixing $|V_{\ell N}|^{2}$
at the high-energy hadron colliders (see Fig. 14). For instance, for the heavy
neutrino mass $m_{N}=200$ GeV, the mixing angles $|V_{eN}|^{2}$ and $|V_{\mu
N}|^{2}$ can be probed, respectively, up to $9.3\times 10^{-6}$ and $9.0\times
10^{-6}$ at the future 100 TeV collider. This improves e.g. the sensitivities
of $2.2\times 10^{-4}$ and $2.1\times 10^{-4}$ from Ref. Pascoli:2018heg  with
dynamic jet vetoes. Remarkably, for $m_{N}\lesssim 1$ TeV, the sensitivities
at the 27 TeV and 100 TeV colliders are (much) better than the indirect upper
limits $|V_{eN}|^{2}<2.2\times 10^{-3}$ and $|V_{\mu N}|^{2}<9.0\times
10^{-4}$ from EWPD delAguila:2008pw ; Akhmedov:2013hec ; Antusch:2014woa ;
Blennow:2016jkn .

The rest of the paper is organized as follows: the inverse seesaw is
introduced in Section 2. The hadron collider analysis details are presented in
Section 3, where the Monte Carlo (MC) signal and background generation is
described in Section 3.1, all the distinguishing variables are defined in
Section 3.2, and the ML analysis is performed in Section 3.3. The resultant
sensitivities at the HL-LHC, HE-LHC and the future 100 TeV collider are
obtained in Section 4, before we summarize and conclude in Section 5. Some
details for the longitudinal momentum of neutrino at the hadron colliders are
given in the Appendix.

##  2 Inverse seesaw

The inverse seesaw was proposed in the 1980s in Refs. Mohapatra:1986aw ;
Mohapatra:1986bd . In this model, two sets of fermions $N_{R,\alpha}$ and
$S_{L,\beta}$ (with $\alpha$ and $\beta$ the flavor indices) are introduced,
which are singlets under the SM and carry lepton numbers
$L(N_{R})=L(S_{L})=1$. For simplicity we consider the symmetric case with
three pairs of $N_{R}$ and $S_{L}$. The Yukawa Lagrangian is given by

$\displaystyle-\mathcal{L}_{Y}\ =\ Y_{\alpha\beta}\bar{L}_{\alpha}\Phi
N_{R,\,\beta}+M_{N,\,\alpha\beta}\bar{S}_{L,\,\alpha}N_{R,\,\beta}+\frac{1}{2}\mu_{S,\,\alpha\beta}\bar{S}_{L,\,\alpha}S^{C}_{L,\,\beta}+{\rm
H.c.}\;,$ (1)

where $L_{\alpha}=(\nu_{\alpha},\;\ell_{\alpha})^{\sf T}$ is the SM lepton
doublet, $\Phi$ is the SM Higgs doublet, $S_{L}^{C}\equiv S_{L}^{\sf T}C^{-1}$
is the charge conjugate of $S_{L}$, $M_{N}$ is a Dirac mass term, and
$\mu_{S}$ is the only Majorana mass term. After the electroweak symmetry
breaking, the SM Higgs obtain its nonzero vacuum expectation value
$\langle\Phi\rangle=(0,v/\sqrt{2})^{\sf T}$, which leads to the Dirac mass
term $M_{D}=Yv/\sqrt{2}$. In the flavor basis
$\\{\nu^{C},\;N_{R},\;S^{C}_{L}\\}$, the Lagrangian (1) gives rise to the full
neutrino mass matrix

$\displaystyle{\cal M}_{\nu}\ =\
\begin{pmatrix}{\mathbf{0}}&M_{D}&{\mathbf{0}}\\\ M_{D}^{\sf
T}&{\mathbf{0}}&M_{N}^{\sf T}\\\ {\mathbf{0}}&M_{N}&\mu_{S}\end{pmatrix}\,.$
(2)

In general, $M_{D}$, $M_{N}$, and $\mu_{S}$ are all $3\times 3$ matrices, and
the full mass matrix ${\cal M}_{\nu}$ in Eq. (2) is a $9\times 9$ matrix. In
the limit of $||\mu_{S}||\ll||M_{D}||\ll||M_{N}||$ (with
$||x||\equiv\sqrt{{\rm tr}(x^{\dagger}x)}$ the positive norm of the matrix
$x$), the mass matrix in Eq. (2) can be block diagonalized, which leads to the
light neutrino mass matrix

$\displaystyle M_{\nu}\ =\ M_{D}M_{N}^{-1}\mu_{S}\>(M_{N}^{-1})^{\sf
T}M_{D}^{\sf T}\,.$ (3)

Different from the canonical type-I seesaw, the tiny neutrino masses are
proportional to the small LNV parameter $\mu_{S}$ in the inverse seesaw. In
light of the ’t Hooft’s naturalness criteria tHooft:1979rat , the LNV
parameter $\mu_{S}$ is small, which provides a natural explanation for the
small neutrino masses in the SM. The masses of heavy fermions $N_{R}$ and
$S_{L}$ are determined predominantly by the matrix $M_{N}$. In particular,
they form quasi-Dirac pairs $N_{i}$ and $\bar{N}_{i}$ ($i=1,\,2,\,3$) with
mass eigenvalues at the order of $||M_{N}\mp\mu_{S}/2||$ with a small
splitting of order $||\mu_{S}||$.

In presence of the Majorana mass term $\mu_{R}\bar{N}_{R}N_{R}^{C}$ in the
matrix (2), the active neutrino masses will obtain 1-loop contribution from
the electroweak radiative corrections Dev:2012sg ; Dev:2012bd . Furthermore,
in the limit of $||\mu_{S}||\ll||M_{D}||\ll||M_{N}||\ll||\mu_{R}||$, the heavy
mass eigenstates $N$ are nearly pure Majorana states with mass at the scale of
$||\mu_{R}||$ as in the canonical type-I seesaw. In this case, as a result of
the Majorana nature of $N$, we can expect equal number of lepton number
conserving and LNV events from $N$ decay. For intermediate values of
$\mu_{R}$, i.e. $||\mu_{R}||\sim||M_{N}||$, we can have scenarios with varying
degree of LNV. In particular, the ratio of same-sign dilepton signal
$pp\to\ell_{\alpha}^{\pm}N\ \to\ \ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}W^{\mp}$
to the opposite-sign dilepton signal
$\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}W^{\pm}$ can be anywhere between 0 and
1, depending on the ratio $||\mu_{R}||^{2}/||M_{N}||^{2}$. This will have
important consequences for LNV signals at the high-energy colliders
Dev:2015pga . For the same-sign and opposite-sign dilepton signatures, the SM
backgrounds are different. For simplicity, we will consider here only the
lepton number conserving signals from the heavy pseudo-Dirac neutrinos in the
basic inverse seesaw scenario in Eq. (2).

As seen in Eq. (3), the active neutrino masses are highly suppressed by
$\mu_{S}$, and the Yukawa couplings $Y$ in Eq. (1) are much larger than that
in type-I seesaw with TeV-scale heavy neutrinos. This will enhance the on-
shell production of heavy neutrinos at the LHC and future higher energy
colliders. Following the pseudo-Dirac nature of heavy neutrinos, the same-sign
dilepton signal $\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}W^{\mp}$ at the hadron
colliders is highly suppressed by $\mu_{S}$. For $m_{N}\lesssim 1$ TeV, the
primary signatures of heavy neutrinos at the hadron colliders are from the
trilepton channel Degrande:2016aje ; Willenbrock:1985tj ; Dicus:1991wj ;
Hessler:2014ssa ; Ruiz:2017yyf

$\displaystyle pp$ $\displaystyle\to$
$\displaystyle\ell_{\alpha}^{\pm}\overset{\textbf{(---)}}{N}\ \to\
\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}W^{\pm}\ \to\
\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}\ell_{\gamma}^{\pm}\nu\,.$ (4)

With the $W$ boson decaying leptonically, there are three charged leptons and
significant MET from the neutrino in the final state. For simplicity, we have
assumed that there is only one heavy neutrino involved in the collider
signature analysis below, and all other heavy neutrinos are sufficiently heavy
such that they are irrelevant. The corresponding leading-order parton level
Feynman diagram is shown in the left panel of Figure 1. Although the tiny
neutrino masses are proportional to $\mu_{S}$, the collider signatures in this
paper do not depend on it. For the mass range of $m_{N}\gtrsim 1$ TeV, the
vector-boson fusion (VBF) and gluon fusion processes become the dominant
production channels Datta:1993nm ; Dev:2013wba ; Alva:2014gxa ;
Degrande:2016aje . For example, for the case of $m_{N}=500$ GeV, the cross
section for DY production at the LHC 14 TeV is 63.3 fb when the 0-jet and
1-jet processes are combined, while the cross sections in the gluon fusion and
VBF channels are only 17.9 fb and 15.0 fb, respectively. For simplicity, in
this paper we study the mass range $m_{N}<1$ TeV and focus only on the DY
production mode.

The production in Eq. (4) is mediated by the mixing $V_{\ell N}$ of $N$ with
the active neutrino $\nu_{\ell}$, and we have assumed the heavy neutrino mass
$m_{N}>100$ GeV such that the decay $N\to\ell^{-}W^{+}$ is kinematically
allowed. Then the cross section for the signal in Eq. (4) can be written as

$\displaystyle\sigma(pp\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}\ell_{\gamma}^{\pm}\nu)\simeq\sigma(pp\to\ell_{\alpha}^{\pm}\overset{\textbf{(---)}}{N})\times{\rm
BR}(\overset{\textbf{(---)}}{N}\to\ell_{\beta}^{\mp}W^{\pm})\times{\rm
BR}(W^{\pm}\to\ell_{\gamma}^{\pm}\nu)\,,$ (5)

with ${\rm BR}(\overset{\textbf{(---)}}{N}\to\ell_{\beta}W^{\pm})$ and ${\rm
BR}(W^{\pm}\to\ell_{\gamma}^{\pm}\nu)$ the branching ratios (BRs) for the
decays $\overset{\textbf{(---)}}{N}\to\ell_{\beta}^{\mp}W^{\pm}$ and
$W^{\pm}\to\ell_{\gamma}^{\pm}\nu$, respectively. As the reconstruction of
$\tau$ is highly non-trivial at hadron colliders, we will consider only the
flavors $\alpha,\,\beta,\,\gamma=e,\;\mu$ for the collider analysis in this
paper. The tree-level differential cross section for the subprocess
$qq^{\prime}\to\ell N$ is Datta:1993nm

$\displaystyle\frac{d\hat{\sigma}(qq^{\prime}\to W^{*}\to\ell N)}{d\hat{t}}$
$\displaystyle=$ $\displaystyle\frac{\pi\alpha_{W}^{2}|V_{\ell
N}|^{2}}{12\hat{s}^{2}}\frac{\hat{t}(\hat{t}-m_{N}^{2})}{(\hat{s}-m_{W}^{2})^{2}}\,,$
(6)

where $\hat{s}$ and $\hat{t}$ are Mandelstam variables, $m_{W}$ is the $W$
boson mass, $\alpha_{W}$ is the weak interaction constant defined as
$\alpha_{W}\equiv g^{2}/4\pi$ with $g$ the gauge coupling for the SM gauge
interaction $SU(2)_{L}$. The heavy neutrino $N$ can decay into the SM
particles via the two-body channels $N\to\ell^{-}W^{+}$, $\nu_{\ell}Z$,
$\nu_{\ell}h$, and the corresponding partial decay widths are given by
Buchmuller:1991tu ; Pilaftsis:1991ug

$\displaystyle\Gamma(N\to\ell^{-}W^{+})$ $\displaystyle\ =\ $
$\displaystyle\frac{\alpha_{W}|V_{\ell
N}|^{2}}{16}\frac{m_{N}^{3}}{m_{W}^{2}}\left(1-\frac{m_{W}^{2}}{m_{N}^{2}}\right)^{2}\left(1+\frac{2m_{W}^{2}}{m_{N}^{2}}\right)\,,$
(7) $\displaystyle\Gamma(N\to\nu_{\ell}Z)$ $\displaystyle\ =\ $
$\displaystyle\frac{\alpha_{W}|V_{\ell
N}|^{2}}{32\cos^{2}\theta_{W}}\frac{m_{N}^{3}}{m_{Z}^{2}}\left(1-\frac{m_{Z}^{2}}{m_{N}^{2}}\right)^{2}\left(1+\frac{2m_{Z}^{2}}{m_{N}^{2}}\right)\,,$
(8) $\displaystyle\Gamma(N\to\nu_{\ell}h)$ $\displaystyle\ =\ $
$\displaystyle\frac{\alpha_{W}|V_{\ell
N}|^{2}}{32}\frac{m_{N}^{3}}{m_{W}^{2}}\left(1-\frac{m_{h}^{2}}{m_{N}^{2}}\right)^{2}\,,$
(9)

where $\theta_{W}$ is the Weinberg mixing angle, and $m_{Z}$ and $m_{h}$ are
the masses of the SM $Z$ and Higgs bosons, respectively. With these decay
widths, it is straightforward to calculate ${\rm BR}(N\to l^{-}W^{+})$.
Furthermore, for the parameter space of $m_{N}>100$ GeV and $|V_{\ell
N}|^{2}\gtrsim 10^{-5}$ of interest in this paper (cf. Fig. 14), the
corresponding lifetime of heavy neutrino $\tau_{N}\lesssim 10^{-9}$ cm, making
the heavy neutrino short-lived enough at the high-energy colliders and the
decay products prompt signals.

Figure 1: The trilepton plus MET signal of a heavy Dirac neutrino at the
hadron colliders, without any jet (left panel) or with only one jet (middle
and right panels).

If we have one or two jets in the trilepton events, the production cross
sections at the LHC and future higher energy colliders are also significant
Das:2014jxa ; Das:2015toa ; Degrande:2016aje ; Das:2016hof ; Pascoli:2018rsg ;
Ruiz:2015zca . At the parton level, there are only leptons in the final state
at the leading-order, therefore the extra jets are from initial state
radiation, as shown in the middle and right panels of Fig. 1. In this case,
the $qg$ process is also important, and it turns out that the cross section
for $pp\to N\ell^{+}(\bar{N}\ell^{-})+1j$ is only two to four times smaller
than that for the leading-order process without any jet at the LHC 14 TeV. At
the HE-LHC 27 TeV and future 100 TeV colliders the trilepton events with one
or two jets are more important than at the LHC. More details can be found in
Section 3 below.

##  3 Collider Analysis

In this section we perform a detailed collider analysis on both the pure
leptonic final states of the signature in Eq. (4) and the case with extra jets
at $\sqrt{s}=$ 14 TeV, 27 TeV and 100 TeV. In this study, we take the heavy
neutrino mass range from 100 GeV to 1 TeV, for which the process in Eq. (4) is
the dominant channel.

###  3.1 MC event generation

All the MC event samples are generated using MadGraph5_aMC@NLO v2.8.2
Alwall:2014hca , and then passed to PYTHIA8 Sjostrand:2014zea  for parton
showering, fragmentation, and hadronization. The detector response is
evaluated by using DELPHES 3 deFavereau:2013fsa  with an ATLAS detector card.
In both background and signal processes the parton distribution function (PDF)
data sets NN23LO1 PDF Ball:2012cx ; Ball:2013hta  are chosen, and the MLM
Mangano:2001xp ; Mangano:2006rw  merging/matching procedure is adopted with
xqcut set to be 40 GeV. FastJet Cacciari:2011ma  is integrated by DELPHES for
jet reconstruction, and the anti-$k_{T}$ algorithm Cacciari:2008gp  is adopted
with the parameter R set to be 0.6 and JetPTMin set to be 20 GeV. The decay
tables of heavy neutrino $N$ and the $Z$ and $W$ bosons are computed by
MadGraph5. The decay of $\tau$ lepton generally produces lots of jets in the
detector, which makes the $\tau$ lepton signals suffer a severe contamination
from the huge QCD backgrounds. Therefore, as mentioned above, in this paper we
will consider only electrons and muons in the leptonic final states as our
signal events.

Table 1:  Geometrical acceptance cuts of electrons, muons and jets adopted in our data simulations at $\sqrt{s}=14$ TeV, 27 TeV and 100 TeV. In addition, for the leading lepton we take $p^{\ell,\,{\rm leading}}_{T}>$ 20 GeV. $\sqrt{s}$ (TeV) | electron | muon | jet
---|---|---|---
$p_{T}$ (GeV) | $|\eta|$ |  $p_{T}$ (GeV) | $|\eta|$ |  $p_{T}$ (GeV) | $|\eta|$
14 |  $>$10 |  $<$ 2.47 |  $>$10 |  $<$ 2.7 |  $>$ 20 |  $<$ 4.5
27 |  $>$10 |  $<$ 2.47 |  $>$10 |  $<$ 2.7 |  $>$ 30 |  $<$ 4.5
100 |  $>$15 |  $<$ 2.47 |  $>$15 |  $<$ 2.7 |  $>$ 45 |  $<$ 4.5

At the detector level, the visible physical objects include jets, electrons
and muons for our case, which can be reconstructed by their momenta $p_{T}$,
pseudorapidities $\eta$ and azimuthal angles $\phi$. The MET $E_{T}^{\rm
miss}$ can be reconstructed from all the visible objects via $E_{T}^{\rm
miss}=|-\sum_{v_{i}}{\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}}(v_{i})|$,
with $\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})$ the transverse
momentum of the $i$th visible object. We will use the reconstructed jets,
charged leptons, and MET to perform our signal and background analysis below.
It has been examined that the total inclusive cross sections below are
independent of the matching parameter. More explicitly, at the parton level we
adopt the basic acceptance cuts for both signal and background events in Table
1. At $\sqrt{s}=$14 TeV and 27 TeV, the cuts are the same as those used in the
ATLAS lepton analyses at $\sqrt{s}$ = 13 TeV in Ref. Aaboud:2018jiw . At
$\sqrt{s}=$100 TeV, the $p_{T}$ cuts on charged leptons and jets are
relatively larger than that at 14 TeV and 27 TeV to remove the soft processes.
For concreteness, we set $p_{T}>$ 15 GeV for electrons and muons, and $p_{T}>$
45 GeV for jets. In addition, we require at least one lepton has the
transverse momentum larger than 20 GeV, i.e. $p^{\ell,\,{\rm leading}}_{T}>$
20 GeV. The cuts on $|\eta|$ are the same, assuming that the detector geometry
is identical to ATLAS.

Figure 2: Cross sections of trilepton signal process with $m_{N}=100$ GeV to 1
TeV at $\sqrt{s}=$ 14 TeV (left), 27 TeV (middle) and 100 TeV (right), with
$\ell=\mu$. Exclusive cross sections for the processes with 0 jet (red), 1 jet
(green) and 2 jets (blue) are shown for comparison. The acceptance cuts in
Table 1 have been applied.

The signal is generated based on the model file HeavyN for Dirac neutrinos
Christensen:2008py ; Alloul:2013bka ; Ruiz:2020cjx ; Alva:2014gxa ;
Degrande:2016aje ; Pascoli:2018heg . As stated in Section 2, for simplicity we
have assumed that there is only one heavy neutrino $N$ with mass at the
electroweak scale, while the other two heavy neutrinos are much heavier and
decouple from our analysis at the high-energy colliders. The exclusive cross
sections for the trilepton signal processes in Eq. (4) at $\sqrt{s}=$ 14 TeV,
27 TeV and 100 TeV are shown in the left, middle and right panels of Fig. 2,
respectively, as function of the heavy neutrino mass $m_{N}$. For the sake of
concreteness, we have set the lepton flavor $\ell=\mu$. For the exclusive
processes, we have included also the $pp\to\ell N+j$ and $pp\to\ell N+2j$
processes. As shown in the middle and right panels of Fig. 1, there are not
only $q\bar{q}^{\prime}$ processes, but also $gq$ (or $g\bar{q}$) and $gg$
processes. The exclusive cross sections for signals with 0, 1 and 2 jets are
shown in Fig. 2, respectively, by the red, green and blue curves, and the
acceptance cuts in Table 1 have been applied. It is noteworthy that for a
fixed $m_{N}$ the cross sections for the processes with one jet are roughly
three times smaller than those without any jet. A similar suppression factor
can be observed when the cross sections with two jets are compared with those
with only one jet. Given the exclusive cross sections in Fig. 2, the signal
production cross sections can be modelled well by the first three exclusive
data samples, and it is safe to neglect events with higher jet multiplicity
$n_{j}\geq 3$ in the following study.

Table 2: All possible trilepton final states in charge and flavor spaces, assuming either $|V_{eN}|\neq 0$ or $|V_{\mu N}|\neq 0$.  mixing | trilepton states | signs ($\pm\mp\pm$)
---|---|---
$V_{eN}$ | $eee$ | $e^{\pm}e^{\mp}e^{\pm}$
$ee\mu$ | $e^{\pm}e^{\mp}\mu^{\pm}$
$V_{\mu N}$ | $\mu\mu e$ | $\mu^{\pm}\mu^{\mp}e^{\pm}$
$\mu\mu\mu$ | $\mu^{\pm}\mu^{\mp}\mu^{\pm}$

For simplicity, in the following simulations, we assume there is only either
$V_{eN}\neq 0$ or $V_{\mu N}\neq 0$ in the neutrino sector, which induces the
combinations of charged lepton flavors $eee$ and $ee\mu$ or $\mu\mu e$ and
$\mu\mu\mu$, with the first, second and third one corresponding, respectively,
to the flavor indices $\alpha$, $\beta$ and $\gamma$ in Eq. (4). In the charge
space, the three leptons of an event can be either $+-+$ or $-+-$. All the
resultant trilepton states are tabulated in Table 2. Our trilepton observables
below will be constructed in both the charge and flavor spaces. Note that the
flavor combinations $\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}\ell_{\alpha}^{\pm}$
($\alpha\neq\beta$) imply that we have both nonzero $V_{eN}$ and $V_{\mu N}$,
which is a clear signature of lepton flavor violation (LFV). For instance, the
states $e^{\pm}\mu^{\mp}e^{\pm}$ can originate from the production and decay
chain:

$\displaystyle pp$ $\displaystyle\to$ $\displaystyle
e^{\pm}\overset{\textbf{(---)}}{N}\ \to\ e^{\pm}\mu^{\mp}W^{\pm}\ \to\
e^{\pm}\mu^{\mp}e^{\pm}\nu\,,$ (10)

where the process $pp\to eN$ is induced by the neutrino mixing $V_{eN}$ while
the decay $N\to\mu W$ is from the mixing $V_{\mu N}$. Such a LFV process is
clearly absent in the SM.

The main background events for our signal are from $pp\to ZW^{\pm}$ process in
the SM, where both $Z$ and $W^{\pm}$ bosons decay into leptonic final states,
i.e.

$\displaystyle pp$ $\displaystyle\to$ $\displaystyle ZW^{\pm}\ \to\
\ell_{1}^{\pm}\ell_{2}^{\mp}\ell_{3}^{\pm}\nu\,,$ (11)

with potentially one or two additional jets. The SM backgrounds have the same
particles in the final state as the signal events. To select the signal
events, we demand the following pre-selection conditions for every event:

  * •

The total number of energetic charged leptons is exactly 3, i.e.
$n_{e}+n_{\mu}=3$.

  * •

In order to suppress the background events from the low mass DY process and
hadronic decays, the invariant mass of the two leading leptons should be
larger than 12 GeV, i.e. $m_{\ell_{1}\ell_{2}}$ $>$ 12 GeV.

  * •

The number of jets is not larger than 2, i.e. $n_{j}\leq 2$.

  * •

The missing transverse momentum $E_{T}^{\rm
miss}=|-\sum_{v_{i}}\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$ is
larger than 20 GeV, i.e. $E^{\rm miss}_{T}>20$ GeV.

The SM backgrounds receive also contributions from the off-shell processes
$pp\to Z^{\ast}W,\,\gamma^{\ast}W\to\ell\ell\ell\nu$ Pascoli:2018heg , with
the photon process interfering with the $Z$-boson mediated diagrams. To
estimate the cross section for the off-shell processes and interference terms,
we first obtain the cross section for the purely on-shell
$ZW\to\ell\ell\ell\nu$ process, and then the total cross section for the same
final state, combining all the contributions from the on-shell $ZW$, off-shell
$Z^{\ast}W$, virtual photon $\gamma^{\ast}W$ and the interference terms.
Subtracting the on-shell $ZW$ contribution from the total cross section, we
obtain the off-shell and interference contributions to the backgrounds. As a
demonstration, the cross sections for the on-shell and off-shell plus
interference contributions with $\ell=\mu$ are shown in the second and third
rows of Table 3, where we have applied the geometrical acceptance cuts in
Table 1 and the pre-selection cuts above. As seen in this table, for the 0-jet
and 1-jet processes, the off-shell plus interference contributions are only
roughly 6% and 4% of the cross sections for the corresponding on-shell
processes.

In addition, the four-lepton production $4\ell$ contributes to the
backgrounds, if one of the charged leptons is missed by the detectors
Pascoli:2018heg . However, after the geometrical acceptance cuts and pre-
selection cuts in particular the missing energy cut, such a process can be
effectively removed. As demonstrated in the last row of Table 3, for the 0-jet
and 1-jet cases, the four-lepton process contributes only roughly 5% and 6% to
the backgrounds, respectively. Taking into account the contributions of the
sub-dominant off-shell and four-lepton processes to the SM backgrounds, we
scale the cross sections of the $ZW^{\pm}$ background, e.g. by, respectively,
a factor of 1.11 and 1.10 for the cases of 0-jet and 1-jet with $\ell=\mu$ in
Table 3.

Table 3: Cross sections of the on-shell $ZW$, off-shell plus interference and four-lepton $4\ell$ background processes after geometrical acceptance cuts presented in Table 1 and the pre-selections cuts, with $\ell=\mu$. process | cross section (fb)
---|---
0-jet | 1-jet
$ZW\rightarrow\ell\ell\ell\nu$ | 29.10 | 20.50
$\ell\ell\ell\nu$ | 1.65 | 0.84
(off-shell + interference)
$4\ell$ | 1.56 | 1.25

We use MadGraph5 to generate the matrix elements of the signal processes
$pp\to\ell N$, $pp\to\ell N+j$, and $pp\to\ell N+2j$ and the background
processes $pp\to WZ$, $pp\to WZ+j$, and $pp\to WZ+2j$. The MLM
merging/matching procedure is adopted to produce $10^{4}$ events as an
inclusive data sample, and three values are taken for the jet parameter
$xqcut$, i.e. $xqcut=40$ GeV, 80 GeV, and 100 GeV. As an explicit example, the
cross sections $\sigma/|V_{\ell N}|^{2}$ for the signal of $m_{N}=500$ GeV
with 0, 1 and 2 jets at $\sqrt{s}=14$ TeV for the three $xqcut$ values are
shown in Fig. 3. As clearly seen in this figure, the signal cross sections are
almost the same, differing by $\sim 1\%$ for the $xqcut$ values above. For the
matched cross sections for the signal processes with 0, 1 and 2 jets at
$\sqrt{s}=14$ TeV, 27 TeV and 100 TeV in Fig. 2, we have taken $xqcut=40$ GeV.

Figure 3: Signal cross sections for final states with 0, 1, and 2 jets at
$\sqrt{s}=$ 14 TeV, with $m_{N}=500$ GeV. The parameter $xqcut$ is taken to be
40 GeV, 80 GeV and 100 GeV for comparison.

To demonstrate the task of background and signal event discrimination, we
present in Table 4 the cross sections of signal events with $m_{N}=150$ GeV
and $m_{N}=500$ GeV, and that of the SM background process $pp\to
ZW^{\pm}+\textrm{jets}$ at $\sqrt{s}=$ 14 TeV, 27 TeV and 100 TeV. All the
signal cross sections are normalized to $|V_{\ell N}|^{2}=1$. The exclusive
cross sections for $0$, $1$, and $2$ jets are all presented in Table 4, and we
have applied the acceptance cuts in Table 1. It can be observed in Table 4
that when $m_{N}$ is small, say 150 GeV, the cross sections of signal can be
comparable with that of the SM background processes at $\sqrt{s}=14$ TeV, 27
TeV and 100 TeV. When $m_{N}$ is large, say 500 GeV, the signal cross sections
can be roughly 50 times smaller than that for the SM backgrounds at 14 TeV,
and down to roughly 30 times smaller at the 100 TeV collider.

Table 4:  Exclusive cross sections of trilepton signal process in Eq. (4) with
$m_{N}=150$ GeV or 500 GeV and the dominant SM background in Eq. (11) at
$\sqrt{s}=14$ TeV, 27 TeV and 100 TeV, with 0, 1 or 2 jets. We have taken
$\ell=\mu$ and adopted the acceptance cuts in Tab. 1. Note that the cross
sections for the SM backgrounds with 0 and 1 jet at 14 TeV are different from
those in Table 3, as the cuts are different.

$\sqrt{s}$ (TeV) |  $m_{N}$ = 150 GeV |  $m_{N}$ = 500 GeV | SM background
---|---|---|---
$\sigma_{\rm matched}$/$|V_{\ell N}|^{2}$ (pb) |  $\sigma_{\rm matched}$ (pb)
0-jet | 1-jet | 2-jet | 0-jet | 1-jet | 2-jet | 0-jet | 1-jet | 2-jet
14 | 0.121 | 0.0521 | 0.0172 | 0.00107 | 0.00075 | 0.00032 | 0.0561 | 0.0392 | 0.0214
27 | 0.241 | 0.0947 | 0.0313 | 0.00275 | 0.00184 | 0.000795 | 0.107 | 0.0842 | 0.0475
100 | 0.804 | 0.269 | 0.0923 | 0.0119 | 0.00705 | 0.00321 | 0.346 | 0.327 | 0.183

###  3.2 Feature Observables

In order to enhance the efficiency of discriminating the background and signal
events, in this subsection we define some feature observables. To reconstruct
these observables from charged leptons in the final state, it is crucial to
identify their origins in terms of the background or signal hypothesis. To
this end, we use the background $W^{-}Z$ and the signal $e^{-}N$ as explicit
examples to describe how to identify the origins of charged leptons in our
reconstruction algorithm.

  * •

For the $W^{-}Z\to e^{-}e^{+}e^{-}+E_{T}^{\rm miss}$ background events, it is
known that the charged lepton with positive charge, i.e. $e^{+}$, must come
from the $Z$ boson, labeled as $\ell_{2}^{+}$. To determine the second lepton
$e^{-}$ from $Z$ decay, there is a two-fold ambiguity. In order to choose it
out, we demand that the lepton $\ell_{1}^{-}$ has the minimal
$|m_{\ell_{2}^{+}\ell_{1}^{-}}-M_{Z}|$. Then the last lepton $e^{-}$ is from
$W$ boson decay and can be labelled as $\ell_{3}^{-}$. This would fix the
background kinematics
$W^{-}Z\to\ell_{1}^{-}\ell_{2}^{+}\ell_{3}^{-}\bar{\nu}=e^{-}e^{+}e^{-}\bar{\nu}$.
This can be easily generalized to the cases of
$\mu^{-}\mu^{+}\mu^{-}+E_{T}^{\rm miss}$ and the charge conjugate $W^{+}Z$
background.

  * •

For the background mode $e^{-}e^{+}\mu^{-}+E_{T}^{\rm miss}$, the
determination of kinematics is rather trivial: the opposite-sign same-flavor
lepton pair is from the $Z$ boson decay, while the third one is from the $W$
boson, i.e. $W^{-}Z\to\mu^{-}\bar{\nu}e^{+}e^{-}$. It is equally trivial for
the background $\mu^{-}\mu^{+}e^{-}\bar{\nu}$.

Generally speaking, in the background hypothesis, we need to first reconstruct
the $Z$ boson from a pair of charged leptons $\ell_{1}^{\pm}\ell_{2}^{\mp}$,
and then the kinematic properties of $W$ boson can be reconstructed from the
third charged lepton $\ell_{3}^{\pm}$ and the missing energy. On the other
hand, for the signal hypothesis, the kinematic variables of the heavy neutrino
are related to $\ell_{2}^{\mp}$, $\ell_{3}^{\pm}$ and the missing energy. Note
that DELPHES is unable to handle fakes for charge identification. In the
detector, bremsstrahlung and straight-like tracks produced at high $p_{T}$ are
the main sources of lepton charge mis-identification. The mis-identification
portion is only about 0.3%, which will not significantly affect our results
FernandezPretel:2791516 .

After identifying the origins of these leptons in terms of background and
signal hypotheses, we can reconstruct the feature observables, which are
divided into three categories, i.e. global kinematic observables, observables
for background event vetoing, and observables favoring signal events. Since
there are at least 4 particles in the phase space of our signal and background
events, we need at least 12-dimension phase space, or equally at least 12
independent observables, to describe the collider events. Since the
longitudinal component of the momentum of neutrino in the final state is
missing, we can use the hypothesis that missing energy is from $W$ boson decay
and derive its expected value.

Figure 4: Distributions of global observables $H_{T}$ (left), $E_{T}^{\rm
miss}$ (middle) and $m_{3\ell}$ (right) of SM backgrounds (red) and heavy
neutrino signals with $m_{N}=500$ GeV (blue) and 1000 GeV (green) at
$\sqrt{s}=$ 14 TeV. The distributions are normalised so that their integrals
are equal to 1.

Let us first consider the events without any jet, and list the definitions of
the global kinematic observables for background and signal discrimination. For
illustration purpose, the distributions of SM backgrounds and signals with
$m_{N}=500$ GeV and 1000 GeV at $\sqrt{s}=14$ TeV are presented in Fig. 4 as
the red, blue and green lines, respectively.

  * •

The scalar sum of transverse momenta
$H_{T}=\sum_{v_{i}}|\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$,
where $v_{i}$ runs over all the reconstructed visible particles. This quantity
is related to the mass parameter $m_{N}$ for signal events; for background
events, it related to the transverse momenta of the $W$ and $Z$ bosons. The
distributions for the SM backgrounds and signals are shown in the left panel
of Fig. 4.

  * •

The missing transverse momentum $E_{T}^{\rm
miss}=|-\sum_{v_{i}}\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$.
The corresponding distributions for backgrounds and signals are presented in
the middle panel of Fig. 4.

  * •

The invariant mass $m_{3\ell}$ of trilepton final state. This quantity is
related to the thresholds of the signal and background processes. As seen in
the right panel of Fig. 4, the distributions of $m_{3\ell}$ are significantly
different for the backgrounds and signals.

Figure 5: Distributions of signal related observables $p_{z}^{\nu}$ (upper
left), $\Delta\Theta(\ell_{3},\,\nu)$ (upper middle), $p_{T}^{W}$ (upper
right), $p_{T}^{\ell_{3}}$ (lower left), $m_{T}^{N}$ (lower middle) and
$m_{N}^{\rm rc}$ (lower right) of SM backgrounds (red) and heavy neutrino
signals with $m_{N}=500$ GeV (blue) and 1000 GeV (green) at $\sqrt{s}=$ 14
TeV. As the variable $p_{z}^{\nu}$ is two-folded, the solid lines in the upper
left, upper middle and lower right panels correspond to the solution 0 (larger
absolute value of $p_{z}^{\nu}$) and the dash lines to the solution 1 (smaller
absolute value of $p_{z}^{\nu}$). The distributions are normalised so that
their integrals are equal to 1.

Six feature observables are introduced to describe the phase space in favor of
the signal events, with the distributions for backgrounds and signals shown in
Fig. 5.

  * •

The two-fold longitudinal momentum solutions $p^{\nu,i}_{z}$ (with $i=0,1$) of
the light neutrino in the final state. In our analysis, the transverse
momentum of light neutrino is defined by $p_{T}(\nu)=E^{\rm
miss}_{T}=|-\sum_{v_{i}}\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$,
where $v_{i}$ denotes charged leptons and jets. Since the light neutrino
masses $m_{\nu}\simeq 0$, we have $E^{2}_{\nu}=p^{2}_{T}(\nu)+p^{2}_{z}(\nu)$,
where $E_{\nu}$ and $p_{z}(\nu)$ are the energy and longitudinal momentum of
light neutrino, respectively ATLAS:2012byx . For the hypothesis that light
neutrino is from the on-shell $W$ boson decay, we can have the relation
$(p_{\ell}+p_{\nu})^{2}=m^{2}_{W}$, where $p_{\ell}$ is the four-momentum of
the charged lepton from $W$ decay. Then we can have an quadratic equation for
$p_{z}(\nu)$, which yields two solutions $p_{z}^{\nu,0}$ and $p_{z}^{\nu,1}$,
with $p_{z}^{\nu,0}$ labeling the one with the larger absolute value. More
details can be found in the Appendix. The distributions of $p_{z}^{\nu,0}$ and
$p_{z}^{\nu,1}$ are shown, respectively, as the solid and dashed lines in the
upper left panel of Fig. 5.

  * •

The angular separation $\Delta\Theta(\ell_{3},\nu)$ between the charged lepton
$\ell_{3}$ and the neutrino from $W$ boson decay. As the longitudinal momentum
of neutrino has a two-fold ambiguity, the separation
$\Delta\Theta(\ell_{3},\nu)$ is also two-folded, as seen in the upper middle
panel of Fig. 5. When the heavy neutrino mass $m_{N}$ is large, the daughter
$W$ boson will be highly boosted, and the charged lepton and neutrino from its
decay will be highly collimated. Therefore the separation
$\Delta\Theta(\ell_{3},\nu)$ can be viewed as a measurement of the Lorentz
boost factor of the $W$ boson from $N$ decay.

  * •

The transverse momentum $p_{T}^{W}$ of the reconstructed $W$ boson. The
corresponding distributions are presented in the upper right panel of Fig. 5.
Obviously, when $m_{N}$ is large, $p_{T}^{W}$ tends also to be large.

  * •

The transverse momentum $p_{T}^{\ell_{3}}$ of the lepton $\ell_{3}$
originating from $W$ boson decay. As seen in the lower left panel of Fig. 5,
$p_{T}^{\ell_{3}}$ tends to be large for the signal.

  * •

The transverse mass $m_{T}^{N}$ of the heavy neutrino. The proper mass $m_{N}$
of $N$ can not be fully reconstructed due to the missing information of
longitudinal momentum of the neutrino in the final state. For each event, once
the lepton $\ell_{1}$ is identified, the transverse mass can be defined via

$\displaystyle m_{T}^{N}=\sqrt{2p_{T}^{\ell_{2}\ell_{3}}E_{T}^{\rm
miss}(1-\cos{\Delta\phi({\ell_{2}\ell_{3},E_{T}^{\rm miss}}}))}\,,$ (12)

where $p_{T}^{\ell_{2}\ell_{3}}$ is the transverse momentum of the
$\ell_{2}\ell_{3}$ subsystem, and $\Delta\phi(\ell_{2}\ell_{3},E_{T}^{\rm
miss})$ denotes the azimuthal angle between the $\ell_{2}\ell_{3}$ subsystem
and the MET $E_{T}^{\rm miss}$. The distributions of $m_{T}^{N}$ are shown in
the lower middle panel of Fig. 5.

  * •

Two reconstructed heavy neutrino masses $m_{N}^{{\rm rc},i}$ (with $i=0,\,1$),
which has a two-fold ambiguity as a result of the two solutions
$p_{z}^{\nu,0}$ and $p_{z}^{\nu,1}$. For signal events, one of them must be
close to the real $m_{N}$. While for background events, neither of them is
expected to be condensed in the mass window of $m_{N}$. As seen in the lower
right panel of Fig. 5, $m_{N}^{\rm rc}$ can well distinguish the signal from
backgrounds.

Figure 6: Distributions of background veto observables $m_{\ell_{1}\ell_{2}}$
(upper left), $p_{T}^{\ell_{1}\ell_{2}}$ (upper middle), $\Delta
R(\ell_{1}\ell_{2},\,\ell_{3})$ (upper right), $\Delta R(\ell_{1},\,\ell_{2})$
(lower left), $\Delta y(\ell_{1}\ell_{2},\,\ell_{3})$ (lower middle) and
$p_{T}^{\ell_{2}\ell_{3}}$ (lower right) of SM backgrounds (red) and heavy
neutrino signals with $m_{N}=500$ GeV (blue) and 1000 GeV (green) at
$\sqrt{s}=$ 14 TeV. The distributions are normalised so that their integrals
are equal to 1.

In order to further suppress the SM background events, six more feature
observables are introduced, and the corresponding distributions are shown in
Fig. 6.

  * •

The invariant mass $m_{\ell_{1}\ell_{2}}$ of the two leptons $\ell_{1}$ and
$\ell_{2}$. For background events, $\ell_{1}$ and $\ell_{2}$ are from $Z$
boson decay, and $m_{\ell_{1}\ell_{2}}$ forms a peak at the $Z$ mass $m_{Z}$;
for signal events such an observable is rather broad, as seen in the upper
left panel of Fig. 6. Therefore, this observable should be the main killer for
the background events.

  * •

The transverse momentum $p_{T}^{\ell_{1}\ell_{2}}$ of the $\ell_{1}\ell_{2}$
subsystem. For the background events, it is equivalently the transverse
momentum $p_{T}^{Z}$ of the $Z$ boson. As the $W$ and $Z$ boson masses are
very close, they fly back-to-back in the background events. In other words,
$p_{T}^{W}$ and $p_{T}^{Z}$ are correlated for background events. In contrast,
for the faked $Z$ boson in the signal events, there should be no such a
correlation, and the distribution $p_{T}^{\ell_{1}\ell_{2}}$ is significantly
broader, as shown in the upper middle panel of Fig. 6.

  * •

The angular distance $\Delta R(\ell_{1}\ell_{2},\ell_{3})$ between the
$\ell_{1}\ell_{2}$ subsystem and the charged lepton $\ell_{3}$ from $W$ boson
decay. For the SM backgrounds, when the energies of $W$ and $Z$ bosons are
large, they are Lorentz boosted, which can be measured to some extent by the
separation $\Delta R(\ell_{1}\ell_{2},\ell_{3})$. As seen in the upper right
panel of Fig. 6, this separation has a relatively sharper peak for the signal
events.

  * •

The angular distance $\Delta R(\ell_{1},\ell_{2})$ between the two leptons
$\ell_{1}$ and $\ell_{2}$ reconstructing the $Z$ boson. For background events,
these two leptons are from the $Z$ boson decay. When $Z$ boson is produced
near its peak, most of $Z$ bosons have small boost factors. Then the pair of
leptons from $Z$ decay are expected to fly back-to-back. The signal events
have a relatively broader distribution of $\Delta R(\ell_{1},\ell_{2})$, as
shown in the lower left panel of Fig. 6.

  * •

The rapidity difference $\Delta y(\ell_{1}\ell_{2},\ell_{3})$ between the
$\ell_{1}\ell_{2}$ subsystem and the charged lepton $\ell_{3}$ originating
from $W$ boson. As presented in the lower middle panel of Fig. 6, the signal
events have a relatively smaller $|\Delta y(\ell_{1}\ell_{2},\ell_{3})|$ than
the background events.

  * •

The transverse momentum $p_{T}^{\ell_{2}\ell_{3}}$ of the $\ell_{2}\ell_{3}$
subsystem. As in the signal events these two leptons are from $N$ decay,
$p_{T}^{\ell_{2}\ell_{3}}$ tends to be larger for the signals than for
backgrounds, as seen in the lower right panel of Fig. 6.

Figure 7: Distributions of jet relevant observables $p_{T}^{j}$ (left) and
$\Delta R(j,\,A)$ (right) of SM backgrounds (red) and heavy neutrino signals
with $m_{N}=500$ GeV (blue) and 1000 GeV (green) at $\sqrt{s}=$ 14 TeV. As the
variable $p_{z}^{\nu}$ is two-folded, the solid lines in the right panel
correspond to the solution 0 (larger absolute value of $p_{z}^{\nu}$) and the
dash lines to the solution 1 (smaller absolute value of $p_{z}^{\nu}$). The
distributions are normalised so that their integrals are equal to 1.

For the pre-selected events with one jet, i.e. the 1-jet channel, three more
feature observables are introduced, and their corresponding distributions are
shown in Fig. 7.

  * •

The transverse momentum $p_{T}^{j}$ of jet. The corresponding distributions
for the SM backgrounds and signals are shown in the left panel of Fig. 7.

  * •

The angular distance $\Delta R(j,A)$ between the jet and the rest of the
products, i.e. the three charged leptons and neutrino. As the longitudinal
momentum of neutrino has a two-fold ambiguity, $\Delta R(j,A)$ is also two-
folded, as seen in the right panel of Fig. 7.

Figure 8: Linear correlation coefficient matrix for backgrounds at
$\sqrt{s}=14$ TeV.  Figure 9: Linear correlation coefficient matrix for the
signal with $m_{N}=500$ GeV at $\sqrt{s}=14$ TeV.

The correlations among the variables above can be quantified by their linear
correlation coefficients, which are shown in Figs. 8 and 9, respectively, for
backgrounds and the signal with $m_{N}=500$ GeV at $\sqrt{s}=14$ TeV. It turns
out that most of the coefficients are close to zero. Variable decorrelation
via the square-root of the covariance matrix is the first step of the TMVA
training processes, and ML methods take the correlation matrices into account
during the training processes.

It is very clear in Fig. 7 that the distributions of $p_{T}^{j}$ and $\Delta
R(j,A)$ are rather similar for the SM backgrounds and signal. In other words,
the ML analysis in Section 3.3 will be dominated by the events without any
jet. Since events with jets contribute little to the discrimination of signal
from backgrounds in our analysis and the simulations with two (or more) jets
cost a lot of computing resources, we will not consider the channels with two
(or more) jets from now on. Here we describe the procedure how we perform our
analysis.

  1. 1.

In the MC generation, we produce 0-jet and 1-jet events together. In the
multi-variate analyses, we divide our data sample into two separate subgroups,
i.e. 0-jet and 1-jet data samples.

  2. 2.

For each heavy neutino mass, we perform ML training for the 0-jet and 1-jet
groups, respectively, and obtain the best performance to separate signal and
background events.

  3. 3.

Finally, we combine these two subgroups of data samples to get the prospects
of $|V_{\ell N}|^{2}$ at the high-energy colliders.

Figure 10: Distributions of $m_{\ell_{1}\ell_{2}}$ (left) and $m_{N}^{\rm
rc,0}$ (right) of SM backgrounds (red) and heavy neutrino signals with
$m_{N}=150$ GeV (dashed blue) and 500 GeV (solid blue) at $\sqrt{s}=$ 14 TeV.
The distributions are normalised so that their integrals are equal to 1.

It is noteworthy that the separation power of the variables above varies for
different beam energies and heavy neutrino masses. For example, for most
center-of-mass energies and heavy neutrino mass $m_{N}$, it is found that the
most powerful observable is $m_{\ell_{1}\ell_{2}}$. As shown in the upper left
panel of Fig. 6, the SM background events form a sharp peak near the narrow
window at the $Z$ mass of 91 GeV, while for the signal events, the peak value
of $m_{\ell_{1}\ell_{2}}$ moves in term of the heavy neutrino mass. It is
remarkable in the left panel of Fig. 10 that when $m_{N}$ is around 150 GeV,
the observable $m_{\ell_{1}\ell_{2}}$ for the signal peaks at around 90 GeV,
which makes the separation of signal and background events rather challenging.
It is found that other observables have also less separation power for the
case of $m_{N}=150$ GeV. As an explicit example, the distributions of
$m_{N}^{\rm rc,0}$ for backgrounds and the signal cases $m_{N}=150$ GeV and
500 GeV are shown in the right panel of Fig 10, which are denoted,
respectively, by the red, dashed blue and solid blue lines. It is clear that
for the case of $m_{N}=150$ GeV, the distribution of $m_{N}^{\rm rc,0}$ for
signal almost overlaps with that for the SM backgrounds.

###  3.3 Machine learning methods and analysis

Figure 11: ML estimator distributions of SM backgrounds (red) and heavy
neutrino signal (blue) with $m_{N}=500$ GeV at $\sqrt{s}=$ 14 TeV. The
distributions are normalised so that their integrals are equal to 1. The left
and right panels are for the MLP and BDTG methods, respectively.

As mentioned in the introduction, we will use ML to achieve the optimal
separation power between the background and signal events. In particular, we
adopt two multi-variate analysis methods, i.e. the MLP Fukushima:2013b  and
BDTG 10.2307/2699986  techniques, which have been encoded in the TMVA package
2007physics…3039H . A portion of MC samples are used for ML training. Input
variables are prepossessed for decorrelation via the square-root of the
covariance matrix (cf. Figs. 8 and 9). Nodes are constructed in the ML
algorithms. The number of nodes is related to the number of trees in BDTG,
while that is related to the number of hidden layers and the number of nodes
on each layer in MLP. There are weights connecting those nodes. In ML methods,
more useful variables have larger weights such as $m_{\ell_{1}\ell_{2}}$ and
$m_{N}^{\rm rc}$, while less useful variables have small weights, for instance
$p_{T}^{j}$, $\Delta R(j,A)$, $\Delta R(\ell_{1},\ell_{2})$ and $\Delta
y(\ell_{1}\ell_{2},\ell_{3})$. A loss function is defined to describe the sum
of difference between the machine prediction (the probability to be a signal
event) and data (1 for signal and 0 for background). The goal of training
process is to adjust the weights to minimize the loss function. Since the
weights are free parameters in the ML, the number of weights can not be too
large compared to the sizes of the samples, otherwise the ML will be over-
fitted. In our case, the number of trees is set to be 1000 in BDTG. In MLP
there is one layer, where there are 25 nodes.

These two methods combine the multi-dimensional feature observables in Figs. 4
to 7 into a single likelihood variable. The 0-jet and 1-jet events above are
combined based on their production cross sections (cf. Fig. 2). However, the
jet relevant distributions in Fig. 7 do not contribute much to the separation
power in the ML analysis, as compared to the distributions of charged leptons
and MET in Figs. 4 to 6. The distributions of one-dimensional likelihood
variables in MLP and BDTG for the backgrounds and signal with $m_{N}=500$ GeV
at $\sqrt{s}=14$ TeV are shown in the left and right panels of Fig. 11,
respectively. The background and signal distributions are denoted,
respectively, by the red and blue lines. It is clear that both the two methods
can well distinguish the signal from backgrounds.

Figure 12:  Background rejection versus signal efficiency from BDTG (red) and
MLP (black) at $\sqrt{s}=$ 14 TeV, for the scenarios $m_{N}=150$ GeV (left)
and 500 GeV (right).

After optimizing the ML estimator cuts, we can achieve good background
rejections (i.e. the portion of rejected background w.r.t. the total) and high
efficiencies to pick out signal events (i.e. the portion of signal left w.r.t.
the total). As an explicit example, the rejection curves for the two methods
BDTG and MLP with $m_{N}=500$ GeV are demonstrated in the right panel of Fig.
12, as functions of the signal efficiency. As a comparison, the rejection
curves for the case $m_{N}=150$ GeV are shown in the left panel of Fig. 12. In
both the two panels, the BDTG and MLP methods are depicted in red and black,
respectively. Comparing the two panels of Fig. 12, the separation performance
of the ML estimators is significantly better at $m_{N}=500$ GeV than that at
150 GeV (cf. Fig. 10). For both the two heavy neutrino scenarios in Fig. 12,
the BDTG analysis gives slightly better discrimination performance than MLP,
especially at high signal efficiency. In general, MLP and BDTG methods yield
compatible results. When the number of input observables is the same, the BDTG
method can be about 7.5 times faster than the MLP method in training
processes. Therefore, in this work we will use the results of the BDTG method
and take the MLP method as a cross check.

Figure 13: Background rejection versus signal efficiency for different sets of
variables in MLP at $\sqrt{s}=$ 14 TeV in the 0-jet channel for $m_{N}=150$
GeV (left) and 500 GeV (right). The magenta, red and blue lines are,
respectively, for all the 18 variables in Fig. 4 to 6, the variables
$m_{\ell_{1}\ell_{2}}$, $m_{N}^{\rm rc,0}$ and $m_{N}^{\rm rc,1}$, and the
rest 15 variables.

It is remarkable in Figs. 4 to 6 that the most distinguishing feature
observables are $m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$. To show the
separation power of these observables, the rejection curves for the
observables $m_{\ell_{1}\ell_{2}}$ and the $m_{N}^{\rm rc}$, the other 15
observables in Figs. 4 to 6 and all the 18 observables are shown in Fig. 13,
respectively, as the red, blue and magenta lines. The left and right panels
are for the cases of $m_{N}=150$ GeV and 500 GeV, respectively. As clearly
seen in Fig. 13, the discrimination power is dominated by
$m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$ for both the two scenarios. For
the challenging case $m_{N}=150$ GeV, including the other 15 observables can
undoubtedly improve the yield, where the distributions of
$m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$ overlap for the signal and
backgrounds (cf. Fig. 10).

Compared to the cut-based analysis, the ML methods can not only use the full
information of the distributions in Figs. 4 to 7, but also take into account
the correlations among the variables (cf. Figs. 8 and 9). Optimizing the
discrimination power of the signal and background data sets, the ML analysis
can significantly improve the prospects of $|V_{\ell N}|^{2}$ at the future
high-energy hadron colliders. As seen in Fig. 14 below, the mixing angles
$|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ can be probed down to ${\cal O}(10^{-6})$
to ${\cal O}(10^{-5})$ for $m_{N}=100$ GeV at the future hadron colliders,
which is roughly one order of magnitude better than the current sensitivities
in the literature.

##  4 Sensitivities of $|V_{\ell N}|^{2}$ at future hadron colliders

The prospects of the heavy-light neutrino mixing parameter $|V_{\ell N}|^{2}$
at the high-energy $pp$ colliders depend on the heavy neutrino mass $m_{N}$
and the cut value of ML estimators as well as the colliding energy and total
luminosity. For a given collision energy and heavy neutrino mass $m_{N}$,
after setting the ML estimator cuts, the number of background ($B$) and signal
($S$) events and the resultant significance $S/\sqrt{S+B}$ can be computed. A
genetic algorithm GATOR Luo:2019qbk  is used to scan the cut values of the ML
estimators to find the $95\%$ C.L. sensitivities of $|V_{\ell N}|^{2}$ with
$S/\sqrt{S+B}=2$. We have chosen three typical collision energies and
integrated luminosities, i.e. the HL-LHC 14 TeV with an integrated luminosity
of 3 ab-1, the HE-LHC with $\sqrt{s}=27$ TeV and 15 ab-1, and the future 100
TeV collider with 30 ab-1.

The prospects of $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ at the high-energy
hadron colliders are shown in the upper and lower panels of Fig. 14,
respectively, as function of the heavy neutrino mass $m_{N}$. Next-to-leading
order corrections have been taken into consideration: the cross section of
signal is enlarged by a factor of 16% Das:2016hof ; Degrande:2016aje ;
Pascoli:2018heg , while that for the SM backgrounds by a factor of 30%,
including the contributions from the additional off-shell and $4\ell$
processes PhysRevD.50.1931 ; Campanario:2010hp ; Pascoli:2018heg . The
corresponding sensitivities of $|V_{\ell N}|^{2}$ at the HL-LHC, HE-LHC and
future 100 TeV collider are shown as the darker yellow, blue and purple lines
in Fig. 14, respectively. We have also estimated the systematic errors, which
include the uncertainties in the cross sections due to MC statistics, scale
variation and the PDF variation for both backgrounds and signal. It is found
that the systematic error associated with the MC statistics is less than 5%
over the entire heavy neutrino mass range in this paper. The uncertainties in
the cross sections due to scale variation and PDF variation are at the order
of 10% and $2\%$, respectively. The resultant systematic uncertainties of
$|V_{\ell N}|^{2}$ are $\lesssim$ 10%. The effects of systematic uncertainties
on the sensitivities of $|V_{\ell N}|^{2}$ are indicated by the lighter color
bands in Fig. 14. As implied by the distributions in Fig. 10, the
sensitivities at $m_{N}\sim 150$ GeV are relatively weak.

Figure 14: Sensitivities of the heavy-light neutrino mixing $|V_{eN}|^{2}$
(upper) and $|V_{\mu N}|^{2}$ (lower) at 95% C.L. as function of heavy
neutrino mass $m_{N}$ from the ML analysis of the trilepton signal in Eq. (4).
The solid darker yellow, blue and purple lines correspond, respectively, to
the central values of sensitivities at the HL-LHC 14 TeV with a luminosity of
3 ab-1, HE-LHC 27 TeV with 15 ab-1 and the 100 TeV collider with 30 ab-1. The
lighter color bands are due to the systematic uncertainties. The dashed lines
are the corresponding sensitivities using dynamic jet vetoes from Ref.
Pascoli:2018heg . The dashed black line in the upper panel is the sensitivity
at the ILC 500 GeV with 500 fb-1. The shaded regions are excluded by the
direct searches of $N$ by L3 (orange) L3:1999ymc ; L3:2001zfe , CMS (brown)
CMS:2018iaf , precision SM Higgs data (green) Das:2017zjc , and the indirect
limits from EWPD (magenta) delAguila:2008pw ; Akhmedov:2013hec ;
Antusch:2014woa ; Blennow:2016jkn . See text for more details.

There are a large variety of laboratory, astrophysical and cosmological limits
on the mixing of heavy neutrinos with the active neutrinos. See Refs.
Deppisch:2015qwa ; Bolton:2019pcu  for a complete list for current
experimental limits on $|V_{eN}|$ and $|V_{\mu N}|$. When the heavy neutrino
mass is larger than roughly 100 GeV, the constraints are mostly from the
direct searches of $N$ at the LEP and LHC, the precision Higgs data and EWPD.

  * •

The heavy neutrinos have been searched for at the LEP by the L3 collaboration
via the process $e^{+}e^{-}\to N_{\alpha}\nu_{\alpha}$ L3:1999ymc ; L3:2001zfe
, with heavy neutrino mass up to 205 GeV. For a 100 GeV $N$, the mixing
$|V_{eN}|^{2}$ can be constrained to the order of 0.002, and is shown as the
orange shaded region in the upper panel of Fig. 14. There are also searches of
heavy neutrinos from $Z$ boson decay via $e^{+}e^{-}\to Z\to\nu N$ for $m_{N}$
below the $Z$ boson mass by L3 L3:1990aqk ; L3:1992xaz , DELPHI DELPHI:1991yjf
; DELPHI:1996qcc  and OPAL OPAL:1990fcc . These limits are not shown in Fig.
14.

  * •

The searches of heavy Majorana neutrinos in the LNV signals have been
performed by ATLAS ATLAS:2015gtp ; ATLAS:2019kpx  and CMS CMS:2012wqj ;
CMS:2015qur ; CMS:2016aro ; CMS:2018szz ; CMS:2018jxx . However, the LHC
searches of LNV signals are not applicable to the Dirac neutrinos in this
paper. To set limits on the Dirac $N$, we adopt the limits from the CMS data
CMS:2018iaf , which utilize all the possible combinations of electrons and
muons in the process $pp\to W^{(\ast)}\to\ell\ell\ell\nu$ including both LNV
and lepton number conserving signals from Majorana neutrinos. In absence of
signal, these data can be interpreted as conservative limits on a Dirac
neutrino $N$. The CMS limits on $m_{N}$ and $|V_{\ell N}|^{2}$ (with
$\ell=e,\,\mu$) are presented as the brown shaded regions in Fig. 14.

  * •

In the inverse seesaw, the heavy neutrino couples to the SM Higgs doublet via
the first term in Eq. (1). When the heavy neutrino is lighter than the SM
Higgs boson, it can be produced from SM Higgs via the exotic decay $h\to\nu
N\to\ell\ell\nu\nu$ Das:2017zjc , which has the same final states as the $h\to
WW^{\ast}$ channel. Therefore the neutrino mixing angle $|V_{\ell N}|$ can be
constrained by the Higgs $WW^{\ast}$ data. Adopting the LHC 8 TeV data
ATLAS:2014aga , the limits on $m_{N}$ and $|V_{\ell N}|$ in Ref. Das:2017zjc
are shown as the green shaded regions in Fig. 14, which exclude the heavy-
light neutrino mixing $|V_{\ell N}|^{2}$ up to ${\cal O}(10^{-3})$ for a 100
GeV heavy neutrino.

  * •

Through the heavy-light neutrino mixing $V_{\ell N}$, the heavy neutrino
couples to the $W$ boson, which induces the double-beta collider process
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{\pm}$ if $N$ is a Majorana fermion
Fuks:2020att . In analogous to the case of neutrinoless double-beta decays,
the heavy neutrino plays the role of mediator in the double-beta collider
process, and can not be produced directly at colliders. It is found that at
HL-LHC 13 TeV with a luminosity of 3 ab-1, the mixing angle $|V_{\mu N}|^{2}$
can be probed down to 0.03 Fuks:2020att . However, such a process is not
relevant to the Dirac heavy neutrino in this paper.

  * •

The heavy-light neutrino mixing will induce the non-unitarity of light
neutrino mixing matrix $V_{\rm PMNS}$, and thus has effects on the EW
precision observables, such as the measurements of weak mixing angle, the $W$
boson mass, the precise $W$ and $Z$ boson decay data, the lepton universality
tests in the meson, tau and $W$ boson decays and the tests of unitarity of CKM
matrix delAguila:2008pw ; Akhmedov:2013hec ; Antusch:2014woa ; Blennow:2016jkn
. It turns out that the EWPD exclude the mixing angle $|V_{eN}|^{2}>2.2\times
10^{-3}$ and $|V_{\mu N}|^{2}>9.0\times 10^{-4}$, as indicated by the magenta
shaded regions in Fig. 14.

There have been intensive studies of the searches of heavy neutrinos at the
high-energy lepton colliders. For instance, the heavy neutrino can be searched
in the channel $e^{+}e^{-}\to\nu N$ at the ILC, which is mediated by a $Z$
boson in the $s$-channel or a $W$ boson in the $t$-channel. With
$\sqrt{s}=500$ GeV and an integrated luminosity of 500 fb-1, the mixing angle
$|V_{eN}|^{2}$ can be probed up to the order of $10^{-4}$ for $m_{N}\lesssim
400$ GeV Banerjee:2015gca , as indicated by the dashed black line in the upper
panel of Fig. 14. At the CLIC with $\sqrt{s}=1.5$ TeV and 3 TeV, the heavy
neutrino can be probed to a larger mass, but the decay products from heavy
neutrino will be more collimated and form fat jets; more details can be found
e.g. in Refs. Chakraborty:2018khw ; Das:2018usr .

The hadron collider searches of heavy neutrinos in the trilepton signal can be
found e.g. in Refs. Das:2014jxa ; Das:2015toa ; Das:2016hof ; Antusch:2016ejd
; Das:2018hph ; Pascoli:2018heg ; Pascoli:2018rsg , and it is found that in
this channel the relative amounts of leptonic and hadronic activities on an
event-by-event basis are very useful to suppress the SM backgrounds
Pascoli:2018heg . Considering both the trilepton process in Eq. (4) and the
VBF $W^{\ast}\gamma$ process which also generates three charged leptons plus
MET and jets, the prospects of $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ from Ref.
Pascoli:2018heg  are shown as the dashed lines in the upper and lower panels
of Fig. 14, respectively. The dashed yellow, blue and purple lines are,
respectively, for the HL-LHC with $\sqrt{s}=14$ TeV and the luminosity of 3
ab-1, HE-LHC 27 TeV with 15 ab-1 and 100 TeV collider with 30 ab-1. As seen in
Fig. 14, ML can improve significantly the prospects of heavy-light neutrino
mixing angles $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ for heavy neutrino mass
$m_{N}\lesssim 1$ TeV in the process in Eq. (4), by up to roughly one order of
magnitude. In particular, for $m_{N}=100$ GeV, at the HL-LHC 14 TeV with a
luminosity of 3 ab-1, the heavy-light neutrino mixing angle $|V_{eN}|^{2}$ and
$|V_{\mu N}|^{2}$ can be probed down to $(2.01_{-0.24}^{+0.16})\times 10^{-5}$
and $(1.53_{-0.18}^{+0.18})\times 10^{-5}$, respectively. At the HE-LHC 27 TeV
with 15 ab-1, the sensitivities of $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ can be
improved up to $(6.48_{-0.62}^{+0.83})\times 10^{-6}$ and
$(5.28_{-0.48}^{+0.64})\times 10^{-6}$. At the future 100 TeV collider, the
prospects are $(3.96_{-0.30}^{+2.18})\times 10^{-6}$ for $|V_{eN}|^{2}$ and
$(2.75_{-0.54}^{+0.85})\times 10^{-6}$ for $|V_{\mu N}|^{2}$, respectively.
For the large mass $m_{N}=1$ TeV, the future 100 TeV collider can even probe
the mixing angles $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ down to
$(2.58_{-0.33}^{+0.61})\times 10^{-4}$ and $(2.68_{-0.42}^{+0.36})\times
10^{-4}$, well below the current EWPD limits.

##  5 Conclusion

Emerging from many different variations of the seesaw framework, the heavy
neutrinos have been one of the well-motivated BSM particles at the high-energy
colliders. Depending on the seesaw model details, the heavy neutrinos can be
either Dirac or Majorana fermions, or a mixture of them. In this paper, we
have studied the application of machine learning to trilepton signals of Dirac
heavy neutrino at the high-energy hadron colliders, which can arise for
instance in the inverse seesaw model. To be specific, we focus on the charged-
current DY process in Eq. (4), which is mediated by the SM $W$ boson and
heavy-light neutrino mixing angles. This process is the dominant production
channel at the hadron colliders for heavy neutrino mass $m_{N}\lesssim 1$ TeV.

With up to one QCD jet, all the kinematic distributions of the charged
leptons, missing energy and jets can be found in Figs. 4 to 7. It turns out
that the observables $m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$ are the most
important distinguishing parameters to suppress the SM backgrounds (cf. Fig.
13). Other observables, like the reconstructed $W$ boson etc., can also
improve the sensitivities especially for $m_{N}\sim 150$ GeV, where the
distributions of observables of the signal events overlap largely with those
of the SM events (cf. Figs. 10 and 13). The distributions of one-dimensional
likelihood variables for the SM backgrounds and signal are exemplified in Fig.
11, and the resultant sensitivities of $m_{N}$ and $|V_{\ell N}|^{2}$ (with
$\ell=e,\,\mu$) at the 95% C.L. using machine learning are shown in Fig. 14,
assuming the heavy neutrino mixes with either $\nu_{e}$ or $\nu_{\mu}$. For
$m_{N}=100$ GeV, the mixing angle $|V_{eN}|^{2}$ ($|V_{\mu N}|^{2}$) can be
probed down to $2.01\times 10^{-5}$ ($1.53\times 10^{-5}$) at the HL-LHC,
$6.48\times 10^{-6}$ ($5.28\times 10^{-6}$) at the HE-LHC, and $3.96\times
10^{-6}$ ($2.75\times 10^{-6}$) at the 100 TeV collider, respectively. For
$m_{N}=1$ TeV, the mixing angles $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ can be
probed, respectively, down to $2.58\times 10^{-4}$ and $2.68\times 10^{-4}$ at
the future 100 TeV collider. This improves significantly the prospects of
heavy neutrinos in the trilepton channel at the high-energy hadron colliders.

This paper has been focused only on the case of heavy Dirac neutrinos in the
specific channel (4). Machine learning can also be applied to other production
channels such as the gluon fusion and VBF processes, which are important for
the heavy neutrino mass above roughly 1 TeV. For Majorana heavy neutrinos,
there will be LNV same-sign dileptons in the final state. For such signals,
the SM backgrounds will be different from those in this paper, and machine
learning may also help to improve the prospects of LNV signals at future high-
energy colliders. These cases will be pursued in future publications.

## Acknowledgements

J.F., Y.P.Z. and H.H.Z. are supported in part by the National Natural Science
Foundation of China under Grant No. 11875327, the Fundamental Research Funds
for the Central Universities, and the Sun Yat-Sen University Science
Foundation. Q.S.Y. is supported by the National Natural Science Foundation of
China under Grant No. 11875260. The work of Y.Z. is supported by the National
Natural Science Foundation of China under Grant No. 12175039, the 2021 Jiangsu
Shuangchuang (Mass Innovation and Entrepreneurship) Talent Program No.
JSSCBS20210144, and the “Fundamental Research Funds for the Central
Universities”.

##  Appendix A Two solutions for the longitudinal momentum of neutrino

Considering the process $W^{+}\xrightarrow{}\ell^{+}\nu$ at the hadron
colliders, we define the momenta of $\ell^{+}$ and $\nu$ as $p_{\ell}$ and
$p_{\nu}$, respectively. As there might also be additional jets in our
analysis, we use $p_{\rm vis}$ to denote the sum of momenta of all the visible
particles, including charged leptons and jets in our case, i.e.

$\displaystyle p_{\ell}$ $\displaystyle\ =\ $
$\displaystyle(E_{1},p_{x_{1}},p_{y_{1}},p_{z_{1}}),$ (13) $\displaystyle
p_{\rm vis}$ $\displaystyle\ =\ $
$\displaystyle(E_{2},p_{x_{2}},p_{y_{2}},p_{z_{2}}),$ (14) $\displaystyle
p_{\nu}$ $\displaystyle\ =\ $
$\displaystyle(\sqrt{p_{x_{2}}^{2}+p_{y_{2}}^{2}+(p_{z}^{\nu})^{2}},-p_{x_{2}},-p_{y_{2}},p_{z}^{\nu}).$
(15)

For on-shell $W$ boson decay, the $W$ boson mass $m_{W}$ satisfies
$m_{W}^{2}=(p_{\ell}+p_{\nu})^{2}$, from which we obtain a quadratic equation
for the longitudinal momentum $p_{z}^{\nu}$ of neutrino:

$\displaystyle a(p_{z}^{\nu})^{2}+bp_{z}^{\nu}+c=0,$ (16)

where,

$\displaystyle a$ $\displaystyle\ =\ $ $\displaystyle
4p_{z_{1}}^{2}-4E_{1}^{2}\,,$ (17) $\displaystyle b$ $\displaystyle\ =\ $
$\displaystyle 4c_{0}p_{z_{1}}\,,$ (18) $\displaystyle c$ $\displaystyle\ =\ $
$\displaystyle c_{0}^{2}-4E_{1}^{2}(p_{x_{2}}^{2}+p_{y_{2}}^{2})\,,$ (19)

with

$\displaystyle c_{0}\ =\
m_{W}^{2}-p_{\ell}^{2}-2p_{x_{1}}p_{x_{2}}-2p_{y_{1}}p_{y_{2}}\,.$ (20)

When $b^{2}-4ac\geq 0$ the equation (16) has two real solutions. The solution
with the larger absolute value is labelled as $p_{z}^{\nu,0}$ and the other
one as $p_{z}^{\nu,1}$. If $b^{2}-4ac<0$, the two solutions will be complex.
Then we will keep only the real parts.

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