# Left-Right Symmetry at FCC-hh

Miha Nemevšek<EMAIL_ADDRESS>Jožef Stefan Institute, Jamova 39, 1000
Ljubljana, Slovenia Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Fabrizio Nesti
<EMAIL_ADDRESS>Dipartimento di Scienze Fisiche e Chimiche,
Università dell’Aquila, via Vetoio, I-67100, L’Aquila, Italy INFN, Laboratori
Nazionali del Gran Sasso, I-67100 Assergi (AQ), Italy

###### Abstract

We study the production of right-handed $W_{R}$ bosons and heavy neutrinos $N$
at a future 100 TeV high energy hadron collider in the context of Left-Right
symmetry, including the effects of $W_{L}-W_{R}$ gauge-boson mixing. We
estimate the collider reach for up to 3/ab integrated luminosity using a
multi-binned sensitivity measure. In the Keung-Senjanović and missing energy
channels, the 3$\,\sigma$ sensitivity extends up to $M_{W_{R}}=35$ and 37 TeV,
respectively. We further clarify the interplay between the missing energy
channel and the (expected) limits from neutrinoless double beta decay
searches, Big Bang nucleosynthesis and dark matter.

###### pacs:

12.60.Cn, 14.70.Pw, 11.30.Er, 11.30.Fs

##  I Introduction

With the enduring experimental successes of the Standard Model (SM), it is
striking that we still lack a definitive theory of neutrino masses. A hint for
going beyond the SM might be found in its structure, where the fermion quantum
numbers seem to point to an underlying parity symmetric theory. This is in
sharp contrast with the maximal breaking of parity observed in the weak
sector. This clash was resolved in the Left-Right (LR) symmetric theories
Pati:1974yy ; Mohapatra:1974hk ; Senjanovic:1975rk ; Senjanovic:1978ev ;
Minkowski:1977sc ; Mohapatra:1979ia and turned out to be deeply connected with
the issue of neutrino mass origin.

In the minimal LR symmetric model (LRSM) parity is broken spontaneously
Senjanovic:1975rk ; Senjanovic:1978ev ; Minkowski:1977sc ; Mohapatra:1979ia ,
together with the new right-handed (RH) weak gauge group $SU(2)_{R}$. The
fermion sector then keeps the parity symmetry, while the gauge sector does
not. Spontaneous symmetry breaking is triggered by a $SU(2)_{R}$-triplet
scalar $\Delta_{R}$ that simultaneously generates the masses of additional
gauge bosons $W_{R}$ and $Z_{LR}$, as well as the masses for RH neutrinos $N$.
Their masses mainly come from a Majorana type Yukawa term that generates the
$N$ mass and breaks the total lepton number after $\Delta_{R}$ gets a vacuum
expectation value (VEV). The residual SM gauge group is then finally broken
via a LR bi-doublet scalar field, which contains the SM Higgs doublet $h$ and
an extra heavier doublet $H$. The bi-doublet has two VEVs that may give rise
to a mixing of the SM $W$ and $W_{R}$. After the completion of electroweak
breaking, light neutrinos also get their Majorana masses with contributions
from the celebrated see-saw mechanism Minkowski:1977sc ; Mohapatra:1979ia ;
Sawada:1979dis ; Glashow:1979nm ; Gell-Mann:1979vob .

In general, to uncover the true microscopic picture of particle mass origin,
we need to perform direct searches at colliders and measure the masses and
couplings of elementary particles, just like we did with the Higgs boson.
Neutrinos are no exception and ultimately we would need to make a direct
discovery at high energy colliders to solidify our understanding of their mass
origin. Only such machines would allow us to perform direct searches for
resonances, such as the $W_{R}$, and give us immediate access to heavy
Majorana neutrinos $N$. In the golden Keung-Senjanović (KS) process
Keung:1983uu , the $W_{R}$ is Drell-Yan produced and decays into a right-
handed (RH) charged lepton $\ell_{R}$ and $N$. In turn, $N$ decays dominantly
through a possibly off-shell $W_{R}$ into another lepton and two jets with the
exact signal depending on its mass $m_{N}$ Ferrari:2000sp ; Nemevsek:2011hz ,
see FIG. 1. Owing to the Majorana nature of $N$, the two leptons have the same
electromagnetic charge half of the times, revealing the breaking of lepton
number (see Gluza:2016qqv for departures from pure Majorana states). Lighter
$N$ becomes boosted and its decay products collimate into a single neutrino
jet Mitra:2016kov . Finally, if $N$ is below $\sim 100\,\text{GeV}$ it becomes
long lived and manifests itself as a lepton plus missing energy $\ell\not{E}$
signature Nemevsek:2011hz . The current LHC searches cover the range of well
separated objects in $\ell\ell jj$ ATLAS:2018dcj ; CMS:2021dzb ; ATLAS:2023cjo
, the collimated “neutrino” jets CMS:2021dzb ; ATLAS:2019isd ; ATLAS:2023cjo
or as a lepton with missing energy CMS:2022krd .

Figure 1: Final states of the $W_{R}$ production, with a prompt charged lepton
plus $N$ decay products ranging from: lepton and jets (KS, left), the prompt
merged $j_{N}$ (center) and the missing energy (right).

The above $W_{R}$ searches loose quickly sensitivity when $N$ is progressively
off-shell, $m_{N}>M_{W_{R}}$. In such case, one can resort to neutrino-less
final states, such as di-jets and pairs of SM gauge bosons that appear in the
presence of $W_{L}$–$W_{R}$ mixing. The di-jet resonance searches were
performed in ATLAS:2019fgd ; CMS:2019gwf , the heavy quark final state
$W_{R}\to tb$ was looked for in ATLAS:2021drn ; CMS:2023ldh and the $W_{R}\to
WZ$ channel in CMS:2022pjv . All these limits converge into a current lower
bound on $M_{W_{R}}$ in the range of $4$–$5.7\text{ TeV}$. The expected reach
of the LHC can further extend to $6$–$7\,\text{TeV}$ with large statistics
Nemevsek:2018bbt , so the parameter space accessible by LHC is almost covered.
The aim of this work is to provide a definitive outlook for the 100 TeV
hadronic colliders, connect it to low energy processes and to the physics of
the early universe.

Apart from the existing collider searches, the precision frontier at low
energies also delivers a set of stringent constraints. It was known since the
early days Beall:1981ze that loop-induced flavor changing processes in the $K$
meson sector push the $W_{R}$ scale into the few TeV regime. Moreover, the
LRSM contains an additional doublet $H$ with flavor off-diagonal couplings
that mediate flavor-changing processes even at tree level, which push the LR
scale even higher Senjanovic:1979cta . A number of subsequent works addressed
these issues Ecker:1985vv ; Mohapatra:1983ae ; Zhang:2007fn ; Maiezza:2010ic ;
Bertolini:2012pu . Most recent updates Bertolini:2014sua uncovered the
dominant role of $B$-meson oscillations and set the limit in the ballpark of
$M_{W_{R}}\gtrsim 8\,\text{TeV}$ and $M_{H}\gtrsim 20\,\text{TeV}$. Even if
such $M_{W_{R}}$ may still be marginally or indirectly probed by the LHC, the
heavier $H$ implies that the model has to live at the brink of non-
perturbativity Maiezza:2016bzp ; Maiezza:2016ybz . A heavier $M_{W_{R}}$ would
clearly relax this tension.

In parallel, constraints from CP violation come from the interplay between the
neutron electric dipole moment and meson processes Maiezza:2014ala ;
Bertolini:2019out . These would require $M_{W_{R}}$ to be pushed beyond 10–20
TeV, at least if LR parity $\mathcal{P}$ is adopted, see Maiezza:2021dui for
the discussion of parity as gauge symmetry and the nature of its imposition.
And even if an axion is invoked, CP violation still implies lower bounds in
the ballpark of $10-20\,\text{TeV}$ Bertolini:2020hjc . In case of
$\mathcal{C}$ as LR parity, the additional CP phases are sufficient in order
to accommodate all the CP-violating channels and such bounds go away.

In summary, the LRSM scale is being driven to ever larger scales of
$\mathcal{O}(10)\text{ TeV}$, nearly out of reach of the LHC, but easily
probed by a future 100 TeV hadron collider (FCC). A number of studies have
started addressing this scenario Rizzo:2014xma ; Ng:2015hba ; Dev:2015kca ;
Mohapatra:2019qid , see also CidVidal:2018eel . However, a complete assessment
of the FCC potential for the LRSM, including the simulation of backgrounds and
transitions between different regimes of $m_{N}$, is still missing. In this
work we close this gap and clarify the FCC reach, by taking into account the
standard KS and missing energy channels.

In addition to the usual $W_{R}$ channels, we address the role of the LR gauge
boson mixing $\xi_{LR}$, that leads to an interplay between the production and
decay via the SM $W$. These channels are complemented by those mediated by
Dirac Yukawa couplings that are responsible for the mixing between the light
and heavy Majorana neutrinos. With an input from neutrino oscillations and
masses/mixings of $N$, one can disentangle the seesaw and compute the Dirac
mass matrix for both choices of LR parity: $\mathcal{C}$ Nemevsek:2012iq and
$\mathcal{P}$ Senjanovic:2016vxw ; Senjanovic:2018xtu ; Kiers:2022cyc . We
show that their effect is relevant in the very light RH neutrino mass range.
Here, displaced signatures play a major role Nemevsek:2018bbt and shall be the
subject of dedicated studies, once the detector geometries and efficiencies
are known for the FCC-hh.

Apart from to the involved analyses using displaced vertices, the missing
energy signal can be understood and estimated quite reliably. It is precisely
in this region of parameter space that interesting connections with other
processes arise as well. It turns out that the neutrinoless double beta
($0\nu\beta\beta$) decay rate from $N$ exchange Tello:2010am and from
additional mixed diagrams Barry:2013xxa is able to compete with FCC-hh, given
the (optimistic) sensitivity of forthcoming experiments. Finally, we should
point out the connection to dark matter in the LRSM Bezrukov:2009th ;
Nemevsek:2012cd that may reside in the 20 TeV range Nemevsek:2012cd but is
also subject to additional constraints from large scale structures
Nemevsek:2022anh .

In Section II we review in detail the production of $W_{R}$ and the decay
chains of the RH neutrino $N$. In Section III we discuss the numerical
simulations of relevant backgrounds. In Section IV we analyze the signal
features for the relevant processes, and in Section V we discuss the
assessment of the expected sensitivity and present the results. Section VI
contains the final discussion and in the Appendix A we give more details and
analytic derivations.

##  II Production at the FCC and decay rates

In this section we review the production of $W_{R}$ at a $pp$ collider, its
decay through $N$, including the various $N$ decay channels and the role of
the left-right gauge boson mixing

$\left|\xi_{LR}\right|\simeq\sin
2\beta\left(\frac{M_{W}}{M_{W_{R}}}\right)^{2}\,.$ (1)

Here, $t_{\beta}\equiv\tan\beta=v_{2}/v_{1}$ is the ratio of the two bi-
doublet VEVs, see Maiezza:2010ic for details.

###  II.1 Production of $W_{R}$ …

The production of an on-shell $W_{R}$ proceeds through the Drell-Yan process
involving the two initial partons

$\begin{split}\frac{\text{d}^{2}\sigma_{pp\to
W_{R}^{+}}}{\text{d}x_{1}\text{d}x_{2}}&=\frac{\pi^{2}\alpha_{2}}{N_{c}}\delta\left(\hat{s}-M_{W_{R}}^{2}\right)\times\\\
&\sum_{u,d}|V_{ud}|^{2}\left(f_{u}(x_{1})f_{\overline{d}}(x_{2})+1\leftrightarrow
2\right)\,.\end{split}$ (2)

Here, $x_{1,2}$ are the parton momentum fractions, $\hat{s}=x_{1}x_{2}s$ and
$\sqrt{s}=100\text{ TeV}$ is the center of mass energy, see Appendix A for a
complete derivation.

The above formula also holds when the left-right gauge boson mixing $\xi_{LR}$
is turned on (via $t_{\beta}$) because the contributions from the right and
the left-mixing currents, proportional to $\sin^{2}\xi_{LR}$ and
$\cos^{2}\xi_{LR}$ sum up to one, while the interference terms are suppressed
either by small quark masses or PDFs of the proton.

In Appendix A we collect the rates of the various $W_{R}$ decay channels,
namely dijet, $\ell N$, as well as $WZ/Wh$ mediated by gauge boson mixing. We
find that the parent $W_{R}$ is never produced with a high boost. Indeed, we
find that the maximal boost factor $\gamma$ is given by

$\gamma_{W_{R}}^{\max}\simeq\frac{\sqrt{s}}{2M_{W_{R}}}\,.$ (3)

Moreover, the $W_{R}$ decay products are typically much lighter than
$M_{W_{R}}$, such that they feature back-to-back geometry distinctive of two
body decays. The only relevant exception is the case of $\ell N$ with $N$
being nearly as heavy as $W_{R}$, to be discussed shortly below. For
completeness we report the $k$-factors for the production including NLO
effects in Section IV.

###  II.2 …and $\ell N$

The triple differential cross section for the $pp\to\ell^{+}N$ production via
$W_{R}^{+}$ is given by

$\begin{split}\frac{\text{d}^{3}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\,\text{d}x_{2}\,\text{d}\hat{t}}&=\frac{\pi\alpha_{2}^{2}}{12\,\hat{s}^{2}}\frac{\hat{t}\left(\hat{t}-m_{N}^{2}\right)}{\left(\hat{s}-M_{W_{R}}^{2}\right)^{2}+\left(\Gamma
M_{W_{R}}\right)^{2}}\,\times\\\ &\sum_{ud}\left|V_{ud}V_{\ell
N}\right|^{2}\left(f_{u}(x_{1})f_{\overline{d}}(x_{2})+1\leftrightarrow
2\right)\,,\end{split}$ (4)

where $\Gamma$ is the total decay width of $W_{R}$, and $\hat{s}$ and
$\hat{t}$ are the partonic Mandelstam variables. Integrating over $\hat{t}$
and the PDFs, we get the total cross-section for $pp\to\ell N$

$\begin{split}&\sigma_{pp\to\ell^{+}N}=\frac{\pi\alpha_{2}^{2}}{24N_{c}}\int_{\frac{m_{N}^{2}}{s}}^{1}\text{d}x_{1}\int_{\frac{m_{N}^{2}}{x_{1}s}}^{1}\text{d}x_{2}\\\
&\frac{\hat{s}}{\left(\hat{s}-M_{W_{R}}^{2}\right)^{2}+\left(\Gamma
M_{W_{R}}\right)^{2}}\left(1-\frac{m_{N}^{2}}{\hat{s}}\right)^{2}\left(2+\frac{m_{N}^{2}}{\hat{s}}\right)\\\
&\sum_{ud}\left|V_{ud}V_{\ell
N}\right|^{2}\left(f_{u}\left(x_{1}\right)f_{\overline{d}}\left(x_{2}\right)+1\leftrightarrow
2\right)\,.\end{split}$ (5)

Figure 2:  The total cross-section $pp\to W_{R}\to\ell N$, for fixed
$m_{N}=200\text{ GeV}$ and $t_{\beta}=0$. Blue dots (solid line) represent the
narrow width approximation as calculated by MadGraph (analytically). Light
(dark) blue band corresponds to scale (PDF) variation. The green dots (solid
line) represent the exact $pp\to\ell N$ result via MadGraph (analytics). Green
bands also show the scale and PDF variation, which is smaller for $pp\to
W_{R}\to\ell N$. The gray dashed line shows the $M_{W_{R}}^{-4}$ slope,
normalized to 5 TeV.

We compare the cross sections for $pp\to W_{R}$ from (24) with $pp\to\ell N$
from (5) at $\sqrt{s}=100\text{ TeV}$ and plot them on FIG. 2. These
analytical calculations are shown with white dot-dashed lines and accompanied
by bands that show the uncertainties due to scale variation by a factor of 2,
i.e. $\mu\in[0.5,2]\sqrt{\hat{s}}$. These are compared to the numerical
results from MadGraph, shown with empty dots. They are surrounded by darker
bands showing the uncertainties for scale variation and the lighter bands for
PDF member variation. As a rule of thumb, the cross-sections fall as
$M_{W_{R}}^{-4}$ and this naïve expectation is shown in the gray dashed line.
It is a very good proxy for $W_{R}$ masses up to about a few 10 TeV, above
which the cross-sections go below this simple scaling. The narrow width
approximation in blue does pretty well compared to the exact case of $2\to 2$
scattering (shown in green) but starts to fail at about 20 TeV, missing the
relevant fraction of $W_{R}$ produced off-shell. It also overestimates the
uncertainty in the cross-section due to scale variation and PDF, as the
uncertainty in the exact total cross section stays below 10%.

Figure 3: Leading lepton $p_{T}(\ell_{1})$ distribution (top) and invariant
mass of $\ell N$ (bottom) in the $pp\to\ell N$ production at
$\sqrt{s}=100\text{ TeV}$. Solid lines are $m_{N}=1\text{ TeV}$ while the
dotted, dot-dashed and dashed are 5, 10 and 15 TeV. On the $p_{T}$ plot we
plot the vertical dotted lines of $p_{T}^{\max}(\ell_{1})$ from (6) and color
dashed lines are for $m_{N}=30\text{ GeV}$. In the invariant mass plot below,
the dashed lines correspond to $m_{N}=1\text{ GeV}$.

To better understand the dynamics in the 100 TeV regime, we move from the
total integrated cross-section to kinematical distributions. Let us focus
first on the $p_{T}(\ell)=p_{T}(N)$ distribution of the leading lepton, shown
on the upper frame of FIG. 3 for various $M_{W_{R}}$ and $m_{N}$. One can
distinguish the two regimes of off- and on-shell $W_{R}$ on the left and right
portion of each line. Clearly, the maximal $p_{T}$ of the lepton (and RH
neutrino) is limited by the $W_{R}$ mass and by the center of mass energy via
the PDFs. For $W_{R}$ produced at rest, which is the relevant regime for large
$M_{W_{R}}$, one has

$p_{T}^{\max}(\ell_{1})\simeq\frac{M_{W_{R}}}{2}\left(1-\frac{m_{N}^{2}}{M_{W_{R}}^{2}}\right)\,.$
(6)

This is derived from (51) with an on-shell $W_{R}$, i.e. by setting
$\sqrt{s}=M_{W_{R}}$ and neglecting the masses of charged leptons and protons.
As seen from the upper frame of FIG. 3, where $p_{T}^{\max}(\ell_{1})$ from
(6) is plotted with vertical dotted lines, the maximal $p_{T}$ increases with
$M_{W_{R}}$ when $N$ is light. It starts to decrease when $m_{N}$ goes closer
to the threshold of $M_{W_{R}}$, because $N$ is produced progressively at
rest.

Figure 4: Decay modes of $N$. On the left we show the branching ratios for $N$
decaying to different decay channels. On the right, we plot the $t_{\beta}$
dependence of the total $N$ width in the relevant mass range. The empty
circles refer to the numerical output from MadGraph and the solid lines
correspond to the analytic rates from (7) and (8).

The spectrum at low $p_{T}$ is dominated by the off-shell $W_{R}$ production,
which is increasingly important for heavier $W_{R}$. The available effective
$W_{R}$ invariant mass is limited by the center-of-mass energy via the PDFs to
a few TeV(dashed lines). The cross-section also gets suppressed as $N$ gets
heavier, and together with the lowering of $p_{T}^{\max}(\ell_{1})$ as
described above, one ends up with a single peak in the intermediate region
(blue solid lines on FIG. 3).

When $t_{\beta}$ is turned on, additional $W_{R}$ decays open up, specifically
into $WZ$ and $Wh$, which slightly reduces the branching ratio to $\ell_{R}N$.
On the other hand, the production of $\ell_{R}N$ via $W$ exchange and gauge
boson mixing becomes possible. This leads to an increase of events at the
lower end of the $m_{N}$ spectrum, similar to the off-shell $W_{R}$ case and
is clearly favored for $m_{N}<M_{W}$.

In the lower frame of FIG. 3 we display similar useful distributions of the
total invariant mass $m_{\text{inv}}(\ell N)$ of the $\ell-N$ system. The most
obvious feature is the characteristic peak at $M_{W_{R}}$. Its behaviour at
lower invariant masses is also interesting. For larger $W_{R}$ masses, there
is a significant off-shell plateau at lower invariant masses, see e.g. the
$M_{W_{R}}=34\text{ TeV}$ solid and dashed lines. This is quite sensitive to
the mass of $N$ and is essentially cut off below $m_{N}$, as shown in the
various blue lines for $M_{W_{R}}=28\text{ TeV}$.

Before moving to the decay of $N$, we remark that in the present work we
assume for definiteness that other possible processes in the LRSM do not
interfere with the $W_{R}$ and $N$ production. In particular, given the high
scales involved, one may consider the possibility that the charged components
of the bi-doublet or triplets have a mass in the probed regime. Their effect,
together with the $W_{R}$ channel, considerably complicates the signatures,
due to the number of diverse couplings and mass scales involved. On the other
hand, such studies will become necessary, in case a signal beyond the standard
model will be observed. In the literature, some of these cases were considered
as benchmarks, namely the $H^{\pm}$ or $\Delta_{R}^{\pm\pm}$ production, see
e.g. Barenboim:1996pt ; Huitu:1996su ; Maalampi:2002vx ; Roitgrund:2020cge ;
Mohapatra:2019qid ; Dev:2015kca . Also a dedicated study of $\Delta^{0}$ would
be particularly interesting, since $\Delta^{0}\to NN$ channel can reveal the
spontaneous mass origin of $m_{N}$ and probes lepton number violation in the
Higgs sector Maiezza:2015lza ; Nemevsek:2016enw . Moreover, these channels may
benefit from the large gluon-fusion production cross-section at
$\sqrt{s}=100\text{ TeV}$.

###  II.3 $N$ decay

The RH neutrino $N$ is typically short lived if the LR scales are in the TeV
region. It decays into a secondary charged lepton $\ell_{2}$ and dominantly
via an off-shell $W_{R}$ into two partons, i.e. $N\to\ell q\bar{q}$. Depending
on $m_{N}$ and the resulting boost, the signature varies: from the lepton and
two distinct jets, to the lepton and a single jet, to a single jet including
the lepton. For very low $m_{N}$, the lifetime can be long enough such that
the decay happens at a macroscopical distance within or even outside the
detector, ending up as missing energy, see FIG. 1.

The dominant $N$ decay width is given by

$\begin{split}\Gamma_{N\to\ell^{\pm}q_{i}\bar{q_{j}}}&=2\,\frac{\alpha_{2}^{2}\,m_{N}^{5}}{128\pi
M_{W_{R}}^{4}}\left|V^{\text{\tiny CKM}}_{ij}\right|^{2}\times\\\
&\left(1-8x+8x^{2}-x^{4}-12x^{2}\log x\right)\,,\end{split}$ (7)

where $x=m_{q}^{2}/m_{N}^{2}$ is the heavier quark mass of the two
$m_{q}=\max(m_{i},m_{j})$. In case $m_{N}$ goes below $M_{W}$, the same final
states (with opposite quark chirality) can also be obtained via the standard
$W$ exchange and LR gauge boson mixing, by multiplying (7) with
$\sin(2\beta)^{2}$.

In turn, two body decay channels $N\to\ell W$ open up as soon as
$m_{N}>M_{W}$, both in the presence of LR mixing $\xi_{LR}$ or from the mixing
angle $\theta$ that connects the left and right handed neutrinos via the Dirac
mass term. These two can be grouped together into

$\displaystyle\Gamma_{N\to\ell^{\pm}W^{\mp}}$
$\displaystyle=\frac{\alpha_{2}}{8}m_{N}\left(\theta^{2}+\xi_{LR}^{2}\right)\left(\frac{1}{x}-x\right)\,,$
(8)

with $x=M_{W}^{2}/m_{N}^{2}<1$, where we dropped a small interference term.

Finally, above $m_{N}>M_{W_{R}}$ the $N\to\ell W_{R}$ channel becomes
dominant. It is obtained by the above formula (8) by replacing $M_{W}\to
M_{W_{R}}$ and $\theta^{2}+\xi_{LR}^{2}\to 1$.

Figure 5: Dominant $N$ decay channels as a function of $M_{W_{R}}$, $m_{N}$
and for various choices of $t_{\beta}$. The standard three body KS decay
dominates everywhere for $m_{N}<M_{W_{R}}$, except for $m_{N}$ just above
$M_{W}$, where $N$ can decay to $W\ell$ depending on $t_{\beta}$ via
$\xi_{LR}$, and for very large $M_{W_{R}}$, where decays via the Dirac mass
can be relevant.

The relative weight of the various $N$ decay channels described above can be
understood collectively in a “spaghetti” plot, presented in FIG. 4 (left),
which we exemplify for the case of $M_{W_{R}}=10\text{ TeV}$ and a moderate
value of $s_{2\beta}=0.2$. The presence of the gauge boson mixing allows for
the two-body $N$ decay, starting from $m_{N}\geq M_{W}$, to a few-$100$–1000
GeV, depending on $t_{\beta}$.

On the right frame of FIG. 4 the effect of $t_{\beta}$ on the total $N$ width
can be appreciated for $t_{\beta}=0.05,0.12,0.3$. It is evident that
$t_{\beta}$ only impacts the light $N$ below few-100 GeV. It is worth
recalling here that the allowed vaues of $t_{\beta}$ are different in case the
LR symmetry is chosen to be either $\mathcal{P}$ or $\mathcal{C}$. In the
former case of $\mathcal{P}$ it is limited by flavor bounds near the value
$t_{\beta}\simeq 0.12$ at low $M_{W_{R}}$ Bertolini:2019out , while in the
latter case it is free, up to the perturbativity limit of $t_{\beta}\lesssim
0.5$. In this work we adopt $t_{\beta}=0.3$ as our highest benckmark value.

The dominance of the various decay channels while varying the model parameters
in the $M_{W_{R}}$–$m_{N}$ plane, can be appreciated also in FIG. 5. Here it
is notable that the two-body mixing mediated decays become dominant in a
region just above $M_{W}$, as will be evident in the final results.

The processes mediated by Dirac neutrino masses turn out to be subleading, or
relevant only in the hardly accessible high $M_{W_{R}}$ and displaced $N$
regime. The reason lies in the small magnitude of the Dirac masses, after
recalling that they are in general not free, but predicted by the LRSM model
in connection with the Majorana neutrino mass matrix Nemevsek:2012iq .

In fact the regime of very light $N$ is particularly interesting and
promising. For $m_{N}\lesssim 100\,\text{GeV}$ the decay can be appreciably
displaced from the primary vertex (depending also on the $W_{R}$ mass) while
for even lower $m_{N}$ the decay happen most of the times outside the
detector, appearing as missing energy signature. These regions are naturally
overlapping as we will show below. The displaced decay regime was studied in
Nemevsek:2018bbt for the LHC. For the FCC, the prospects and sensitivities for
observing displaced vertices are strongly dependent on the inner detector
design and vertexing challenges, so that, albeit very interesting, are clearly
premature. The missing energy signature on the other hand are straightforward
and only weakly depend on the detector size. We will analayze it below.

$\begin{array}[]{|c|c|c|c|c|c|c}\hline\cr\qquad\text{Backgrounds
[pb]}&\quad\verb!w+12j!&\verb!DY+12j!&\verb!vv+012j!&\verb!tt+01j!\\\
(\sqrt{s}=100\,{\rm TeV})&\hphantom{\bf 0.0009523}&\hphantom{\bf
0.0009523}&\hphantom{\bf 0.0009523}&\hphantom{\bf 0.0009523}\\\
\hline\cr\texttt{xptj},\texttt{xptl}>50&5700&1000&180&480\\\
\texttt{xptj},\texttt{xptl}>500&4.0&0.45&0.110&0.031\\\
{}+\texttt{xptl}>1000&0.46&0.030&0.017&0.0045\\\
{}+\texttt{misset}<500&0.39&0.030&0.011&0.0028\\\
\hline\cr{}+\texttt{xptj},\texttt{xptl}>1500\;\text{(detector)}&\bf 0.047&\bf
0.0025&\bf 0.001&\bf 0.000012\\\ \hline\cr{}+\text{k-factors}&\bf 0.023&\bf
0.0017&\bf 0.0015&\bf 0.000024\\\ \hline\cr\end{array}$

Table 1:  Table of background cross sections versus various cut requirements.
All the cross sections are in pb and the cuts are in GeV. For each case, the
subprocesses with at least one final lepton were selected. For any jet and
lepton, a minimal momentum is assumed with $\texttt{ptj}>20$,
$\texttt{ptl}>10$. At the analysis level we impose even stronger cuts on the
leading jet/lepton with $\texttt{xptj},\texttt{xptl}>1500$.

##  III Backgrounds

Let us focus on the backgrounds for the cases when $N$ decays inside the
detector (we discuss the missing energy background later on). Given the
signature characteristics, namely at least one highly energetic prompt lepton,
low missing energy, and one or more jets, we identify the following possible
SM processes contributing as backgrounds:

  1. 1.

$W$ boson plus one or two jets;

  2. 2.

Drell Yan $\ell^{+}\ell^{-}$ plus one or two jets;

  3. 3.

diboson production $VV$ with up to two jets.

  4. 4.

$t\bar{t}$ plus up to one jet;

For all of these SM backgrounds we require the presence of at least one
charged lepton. For example, in the $VV$ samples we force at least one of the
bosons to decay leptonically.

All of the backgrounds are generated using MadGraph 3.3.2 and 2.8.0
Alwall:2007st , hadronized using Pythia 8 Sjostrand:2007gs . For detector
simulation we used Delphes 3 deFavereau:2013fsa , adopting the provisional FCC
card FCCcard . The parton level processes are generated at tree level with jet
matching. In addition, a review of the literature for NLO and Electroweak (EW)
corrections brings the following $k$-factors:

  1. 1.

for `w+12j`, NLO$+$EW corrections imply Mangano:2016jyj a striking reduction
of 50% or more, especially at high $p_{T}\gtrsim 10\,\text{TeV}$ as considered
here, so that we apply a 50% reduction;

  2. 2.

for `DY+12j`, the NLO$+\gamma$-induced corrections also bring a reduction of
up to 50%, so that we adopt a correction of $-30\%$;

  3. 3.

for `vv+012j`, the NLO corrections lead to an enhancement of circa 1.5;

  4. 4.

for `tt+01j`, NLO enhances by a factor of $\sim 2.5$ and EW corrections bring
a reduction of 20%, so that an increase of 100% is a safe estimate.

At the same time, the signal is subject to a $k$-factor of 1.2–1.5
Mitra:2016kov for the range of $W_{R}$ masses that we consider here. All of
these estimates are clearly affected by their own uncertainties, motivating
further studies for increasing the precision at 100 TeV.

Figure 6:  Distribution of leading lepton $p_{T}$ before cuts, for the
background (gray) and various signal scenarios,
$M_{W_{R}}=6,10,16,22,28,34\text{ TeV}$ from left to right with a fixed
$m_{N}=1\text{ TeV}$.

As discussed above, after $W_{R}$ decays, the prompt lepton and the subsequent
leading jets typically carry high momentum on the order of $M_{W_{R}}/2$. On
the other hand, backgrounds typically concentrate at lower $p_{T}$s, reaching
at most a few TeV. For illustration we plot the leading lepton $p_{T}$
distributions in FIG. 6. As a result, the sensitivity to the signal can be
efficiently improved by restricting the leading lepton and jet momenta to be
of the order of TeV. At generator level we require both a leading jet and
leading lepton to have $p_{T}>1\text{ TeV}$, using the xptj and xptl
parameters. This reduces the background cross sections substantially, so that
enough statistics can be gathered by Monte Carlo, even for an integrated
luminosity of 3/ab. The $\not{E}<500\text{ GeV}$ cut brings in a further
reduction, without significantly impacting the signal. Finally, we chose to
impose even stronger cuts xptj, $\texttt{xptl}>1500$ at detector level, which
further reduces the first backgrounds by $10$, and the softer `tt+01j` by more
than $10^{3}$.

The efficiency of the above cumulative cuts on the signal varies approximately
from 40% at low $M_{W_{R}}\sim 7\,$TeV to about 90% at highest $M_{W_{R}}\sim
40\,$TeV (the signal is largely off-shell and moves to softer $p_{T}$). The
cut flow and final cross sections for the background are reported in TAB. 1.
The most dominant process is `w+12j`, and the other processes can be safely
neglected in the present study.

As evident from FIG. 6, if one wished to focus solely on the higher $W_{R}$
masses at higher luminosity, then more stringent cuts could even be imposed
from the beginning. This would further reduce the need for large samples in
the generated background. We opt instead for very minimal cuts and a larger
statistics, which should reliably simulate also the lower values of
$M_{W_{R}}$ (still above the reach of the LHC) and lower luminosities. To
assess the sensitivity we employ a cut-free method Nemevsek:2018bbt , as
described in section V.

##  IV Signal

The signal was simulated with the same settings and cuts as the background. We
used the LRSM model file LRSMmix-model at LO, which was introduced in
Roitgrund:2014zka and updated from Maiezza:2015lza .

The single prompt lepton $\ell_{1}$ from $W_{R}$ decay is typically well
isolated and can serve as a high efficiency trigger. At the same time, the $N$
decay products are always very energetic, as shown in the previous sections.
This naturally happens for light $N$ ($m_{N}\ll M_{W_{R}}$), which is boosted
and whose decay products typically have $p_{T}\sim M_{W_{R}}/3$. Likewise, the
heavier $N$ ($m_{N}\lesssim M_{W_{R}}$), features an energetic secondary
lepton and jets that have similarly a large momentum $\sim m_{N}/3$.

Figure 7: Distribution of events in 2$\times$2 TeV bins in the space of
leading lepton $p_{T}(l_{1})$ versus $m_{\text{inv}}(l_{1}j_{1})$. Background
is shown underneath in gray, and overlayed regions correspond to signal
scenarios with $M_{W_{R}}=6,10,16,22,28,34\,$TeV left to right, and fixed
$m_{N}=1$ TeV (for clarity only the highest contours are shown). The signal
moves progressively to a background free region as $M_{W_{R}}$ becomes
heavier.

At the detector level, due to isolation limitations, it is not always possible
to separate all the $N$ decay products, especially when they are boosted and
produce a single jet that contains the secondary charged lepton $\ell_{2}$.
Our approach is thus to reconstruct the $W_{R}$ invariant mass of the
(leading) jet, together with one (leading) lepton, or possibly with two
leptons if they are isolated. The single lepton plus jet variable
$m_{\text{inv}}(l_{1}j_{1})$ is more appropriate for the light (boosted) RH
neutrino regime, while the two lepton plus jet variable
$m_{\text{inv}}(l_{1}l_{2}j_{1})$ is sensitive to the higher RH neutrino
masses. We also always consider leptons and jets with a minimal
$p_{T}(\ell)=10\text{ GeV}$ and $p_{T}(j)=50\text{ GeV}$.

In FIG. 7 we plot the distribution of events in the
$p_{T}(l_{1})$–$m_{\text{inv}}(l_{1}j_{1})$ plane, both for the background and
for a selection of signal scenarios with fixed $m_{N}=1\text{ TeV}$. As one
can see for increasing $M_{W_{R}}$ the signal peaks progressively outside of
the background region. This is characteristic of $s$-channel resonance
searches and is particularly promising.

Varying the RH neutrino mass leads to various distributions that are shown in
FIG. 8. One can observe that because the RH neutrino decays to a further
lepton and jets, and because at detector level one can not distinguish between
leptons, it may happen that the secondary lepton from $N$ decay is harder and
takes the role of the first. This happens at large $m_{N}$ masses, when
$p_{T}(\ell_{2})\simeq m_{N}/3>p_{T}^{\max}(\ell_{1})$ from (6). Solving for
$m_{N}$, we find $m_{N}/M_{W_{R}}=(\sqrt{10}-1)/3\simeq 0.72$, with
$p_{T}^{\max}\simeq M_{W_{R}}/4$. This applies to the rightmost frames in FIG.
8.

Similar distribution of events appear in the
$p_{T}(l_{1})$–$m_{\text{inv}}(l_{1}l_{2}j_{1})$ plane, at large $m_{N}$
masses. In the next section we will take into account both channels and
estimate the sentitivity in the entire $M_{W_{R}}$–$m_{N}$ plane.

Figure 8:  Distribution of leading lepton $p_{T}(l_{1})$ versus
$m_{\text{inv}}(l_{1}j_{1})$, for various heavy neutrino masses, and fixed
$M_{W_{R}}$.

$\begin{array}[]{|lcc|rcccccccccccc|}\hline\cr&\\!\\!\\!\mathcal{L}=3\,\mathrm{ab}^{-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!&&M_{W_{R}}\\!:&10&10&10&10&~{}~{}~{}18&18&18&18&~{}~{}~{}32&32&32&32\\\
\text{variable}&\text{range}&\text{\\#
bins}&m_{N}\\!:&240&1000&3981&9550&~{}~{}~{}240&1000&3981&16596&~{}~{}~{}240&1000&3981&28840\\\
\hline\cr
p_{T}(\ell_{1})&\\{1.5,31.5\\}\,\text{TeV}&{16}&&106&78.5&80.8&2.19&~{}~{}~{}~{}25.8&26.0&25.1&1.21&~{}~{}~{}3.43&3.78&2.73&0.326\\\\[2.15277pt]
m_{\text{inv}}(\ell_{1}j_{1})&\\{1.5,41.5\\}\,\text{TeV}&{20}&&123&89.7&86.9&2.99&~{}~{}~{}29.5&28.2&26.2&1.61&~{}~{}~{}4.06&4.28&3.25&0.531\\\\[2.15277pt]
m_{\text{inv}}(\ell_{1}\ell_{2}j_{1})&\\{1.5,41.5\\}\,\text{TeV}&{20}&&124&94.0&109.&15.7&~{}~{}~{}29.4&28.6&29.6&6.03&~{}~{}~{}4.05&4.32&3.60&0.992\\\\[2.15277pt]
\hline\cr\end{array}$

Table 2:  The first column reports the chosen grid binning variables, their
range and number of corresponding bins. The columns on the right correspond to
sensitivities in $\sigma$ obtained with $3/\,\text{ab}^{-1}$. Subsequent rows
show the progression/optimization of the sensitivity after adding in turn each
binning variable to the grid, and the bottom row represents our final
sensitivity. The selection of points in the $m_{N}-M_{W_{R}}$ parameter space
for progressively large $m_{N}$ ranges from the single lepton to the two
isolated leptons regime.

##  V Sensitivity

While a fair idea of the reach may be obtained from FIG. 6 with a sliding cut
on $p_{T}(\ell)$ as a function of $M_{W_{R}}$, the further dependence on
$m_{N}$ makes this procedure unfeasible. A simpler and optimal method to
assess the sensitivity for any choice of model parameters was devised in
Nemevsek:2018bbt . It consists of splitting the background and signal events
in a multidimentional binning along a few relevant observables, and defining
the overall sensitivity as the sum in quadrature of single bin sensitivities:

$\Sigma^{2}=\sum_{i\in\text{bins}}\frac{s_{i}^{2}}{s_{i}+b_{i}}\,,$ (9)

where $s_{i}$ and $b_{i}$ are the expected number of signal and background
events in each bin (see FIG. 7 for an example of a two-variables binning of
signal and background). The method is quite robust with respect to binning
variations, with a systematic uncertainty that can be suitably controlled. We
refer to Nemevsek:2018bbt for the illustration and the theoretical discussion
of the method.

The KS and LJ signature. The binning grid in this case concerns three
observables: $p_{T}(\ell)$, $m_{\text{inv}}(\ell j)$ and
$m_{\text{inv}}(\ell\ell j)$, which are reported in the first column of TABLE
2. Each of them spans the range (above the minimal cut) in bins of
$2\,\text{TeV}$.

As discussed above, $p_{T}(\ell)$ is already a strong discriminator between
the signal and background. The other two variables address the reconstruction
of the $M_{W_{R}}$ invariant mass, and are useful to boost the sensitivity in
the low ($m_{\text{inv}}(\ell j)$) and heavy ($m_{\text{inv}}(\ell\ell j)$) RH
neutrino mass regimes.

In TABLE 2 we also report the resulting sensitivity for a selection of signal
points in the $M_{W_{R}}$–$m_{N}$ parameter space, where the successive table
rows display the sensitivity obtained by adding in turn the corresponding
variable to the binning. One can notice the increase in sensitivity in the
second line in particular for light $N$, and the strong increase in the last
line, especially for large $N$ masses. In FIG. 9 we display the sensitivity in
the $M_{W_{R}}$–$m_{N}$ plane, for the separate channels where the final
signature has $LLJ$ (left) or $LJ$ (right) only. One can appreciate the
complementarity of the two channels for low and high RH neutrino masses. The
final combined sensitivity is shown in FIG. 10 for an integrated luminosity of
$L=3\text{ ab}$. It is quite notable that the combined reach of the two
channels together is around $37/35/32\text{ TeV}$ at $2/3/5\sigma$ of C.L..
Also for an early run with a low integrated luminosity of $L=30\text{ fb}$,
the 5 $\sigma$ discovery reach is around 15 TeV.

At the lower end of $M_{W_{R}}\lesssim 15\text{ TeV}$, the FCC-hh can probe
also the heavy $m_{N}>M_{W_{R}}$ regime, where $W_{R}$ is produced off-shell,
but with a sufficiently large invariant mass to generate an $N$ with mass up
to $m_{N}\simeq 15\text{ TeV}$.

Figure 9: Sensitivity (number of $\sigma$s) of the resolved two lepton LLJ
(left) and a single lepton LJ (right) channels. Figure 10: The green areas
show the final KS plus LJ sensitivity (in number of $\sigma$s) achievable with
3/ab integrated luminosity. We show also the dependence on
$t_{\beta}=0,0.12,0.3$ (dotted, dashed, long-dashed). The overlayed orange
shaded region in the lower part of the frame displays the 3 and 5 $\sigma$
sensitivity to the $\ell+\not{E}$ signature.

FIG. 10 also shows the impact of the presence of $t_{\beta}$. For
$m_{N}>M_{W}$, we see a depletion of the observed signal rate, as discussed in
section II.3. On the other hand, the production via $W$ opens up below $M_{W}$
and leads to an increase. However, this happens in the light $N$ regime, where
$N$ decay is progressively displaced. Therefore, the dependence on $t_{\beta}$
would be of major interest in future studies of displaced decays at FCC-hh.

The missing energy signature. For light $m_{N}\lesssim 300\,\text{GeV}$, the
$N$ decay length in the lab frame becomes long enough, such that the
probability of it decaying outside of the detector becomes sizeable.
Experimentally this shows up as a prompt lepton plus missing energy, which is
the signature usually assumed in searches for a sequential
$W^{\prime}\to\ell\nu$ CMS:2022krd .

For the FCC-hh we assume a conservative detector size of 5 meters and
calculate the expected number of those events, where $N$ decays entirely
outside of the detector, while $\ell_{1}$ always remains prompt. The technical
details of this analytical calculation are described at the end of Appendix A.
To compare with the estimated expected SM backgrounds, we separate the events
into bins of transverse mass $m_{T}$, as considered in ATLAS:2017jdr ;
CMS:2022krd , but we rescale the cross-sections to $\sqrt{s}=100\text{ TeV}$.
The background turns out to be dominated by single $W$ production and sub-
dominant Drell-Yan, $t\bar{t}$ and multi-jet components.

The final result is shown as a shaded orange region and covers the lower part
of FIG. 10. It demonstrates that in this channel the $3(5)\,\sigma$ expected
FCC reach extends up to $M_{W_{R}}\simeq 37(33)\text{ TeV}$ and up to
$m_{N}\simeq 300\text{ GeV}$. The sensitivity reach here is thus slightly
higher than in the KS and LJ channels, like it already happens at the LHC
Nemevsek:2018bbt , but shows a nice complementarity. The invisible channel
reach estimate would be slightly reduced in case a much larger detector size
will be chosen, but at the advantage of the other channels discussed above. In
addition, the possibility of detecting displaced $N$ decays, not yet
considered at this stage, would boost the sensitivity of KS in the low $m_{N}$
regime, as it was argued in Nemevsek:2018bbt .

An important point concerns the connection between colliders and neutrinoless
double beta decay ($0\nu\beta\beta$) in this light $m_{N}\lesssim 300\text{
GeV}$ regime. There are a couple of new sources for the $0\nu\beta\beta$ rate
present in the LRSM Mohapatra:1979ia ; Mohapatra:1980yp , in addition to the
standard light Majorana neutrino exchange. While the standard double weak
decay produces two outgoing electrons with _left_ chirality, in the LRSM new
diagrams appear with _two right_ or _one left and one right_ -handed outgoing
electron. It turns out that the latter opposite chirality process may be the
increasingly dominant one for heavier $M_{W_{R}}$. It is mediated by the
exchange of two $W$’s, or by one $W$ plus one $W_{R}$. The latter option has a
suppressed nuclear matrix element (see Doi:1985dx ; Hirsch:1996qw ;
Vergados:2002pv and Barry:2013xxa for details), one is left with the double
$W$ exchange that can produce opposite chirality electrons via the LR gauge
boson mixing. This contribution is thus driven by the magnitude of
$t_{\beta}$.

The sensitivity to $0\nu\beta\beta$ experiments is shown in FIG. 11 for a
benchmark value of $t_{\beta}=0.3$ with the calculable111The size of the
mixing is predicted in LRSM via the calculable Dirac mass matrix
Nemevsek:2012iq ; Senjanovic:2016vxw , but might be enhanced in small corners
of parameter space Barry:2013xxa . seesaw Dirac mixing $U_{\nu
N}\simeq(m_{\nu}/m_{N})^{1/2}$. We depict the region where the LRSM can
saturate a possible $0\nu\beta\beta$ evidence, for the present and future
planned sensitivities (GERDA-II, $m_{ee}<92\text{ meV}$, and LEGEND-1000,
$m_{ee}<9-21\text{ meV}$). One can see that the region extends up to the scale
of $M_{W_{R}}\simeq 30\text{ TeV}$, with light $N$ in the (sub)GeV range. For
$t_{\beta}=0.12$ the region extends up to $M_{W_{R}}\sim 15\,\text{TeV}$. We
also recall that $m_{N}<0.14$–few GeV is excluded by the requirement that $N$
decays fast enough, in order not to spoil the BBN predictions Nemevsek:2011aa
; Nemevsek:2012cd , see the gray shading. These come about because the $N$
with such mass is produced thermally in the early universe and then becomes
long-lived with $\tau_{N}\gtrsim 1\text{ sec}$ and therefore spoils the BBN.

Figure 11:  LRSM contribution to $0\nu\beta\beta$ from processes including the
righ-right heavy $N$ exchange and the left-right amplitude via gauge boson
mixing (for a benchmark value of $t_{\beta}=0.3$). The red and blue shaded
regions show the mass ranges saturating various $0\nu\beta\beta$
sensitivities, and their interplay with the invisible $N$ channel at FCC in
orange. The lower part of the diagram in shaded gray shows the exclusion from
BBN.

Finally, this region of parameter space harbors the possibility of having a
(warm) dark matter(DM) candidate. Namely, for very light masses of $N$ close
to the keV scale, one can satisfy the DM abundance with entropy dilution
Bezrukov:2009th ; Nemevsek:2012cd . Indeed, it has been argued in
Nemevsek:2012cd that using the phase space suppression for the dilutor
lifetime and the drop of relativistic degrees of freedom in $g_{*}(T)$ around
the QCD phase transition, an $\mathcal{O}(\text{TeV})$ solution for
$M_{W_{R}}$ may be attainable. This requires a very light $m_{N}$ at about
keV, in line with constraints from Dwarf spheroidals DiPaolo:2017geq . Without
resorting to the $g_{*}$ shift, a second solution for $M_{W_{R}}$ exists in
the $20\text{ TeV}$ range (see FIG. 6 of Nemevsek:2012cd ), which goes further
up for heavier DM candidates (see also Dror:2020jzy for the heavy $M_{W_{R}}$
scenarios).

Recently, it was argued that the possibility of having a DM candidate is
subject to strong constraints from the large scale structure data
Nemevsek:2022anh . This happens because the secondary DM production from
entropy injection spoils the matter power spectrum with potentially
significant impact on the LR scale. The fate of the DM with low $M_{W_{R}}$
remains to be established, but in any case the interplay with future colliders
lies precisely in connecting the DM thermal freeze-out to the missing energy
signal at the FCC-hh.

Thus, on FIG. 11 one can appreciate the interplay between $0\nu\beta\beta$ and
the invisible $N$ channel at FCC: a positive $0\nu\beta\beta$ finding, in
absence of the standard contribution (e.g. because of normal hierarchy, see
the discussion in Dvali:2023snt ) would imply an upper bound on $M_{W_{R}}$
Nemevsek:2011aa that we estimate in the range of $10-30\text{ TeV}$. This
would constitute a case for looking for $W_{R}$ and $N$ at an FCC-hh. Further
experiments at the kton scale are envisaged to push the $0\nu\beta\beta$
search down to an impressive $\sim 1\,\text{meV}$ Avasthi:2021lgy . These
would connect to LRSM scales as high as $M_{W_{R}}\simeq 100\,\text{TeV}$,
even beyond the reach of a 100 TeV collider.

##  VI Outlook

While the quest for a theory of neutrino masses is still open, the LRSM stands
as a unique candidate connecting their origin with an understanding of parity
breaking in weak interactions. Ongoing experimental efforts are at the limit
of their capabilities in probing the model parameter space. This is true for
collider LHC probes, with an estimated reach of $M_{W_{R}}\simeq 7\text{ TeV}$
Nemevsek:2018bbt , and also low energy probes. Most notable are current and
planned $B$-meson flavour observables that will be sensitive to mass scales at
most up to $M_{W_{R}}\simeq 10\text{ TeV}$ Bertolini:2014sua . It is thus
important to address the prospects with the planned and proposed future
experiments, with the energy frontier being the elected arena where to search
for direct signs of new physics. In this work we systematically estimated the
reach of a hadronic ($pp$) FCC with 100 TeV center of mass energy in the
search for a $W_{R}$ decaying leptonically, which has the potential to uncover
lepton number violation, for diverse choices of $M_{W_{R}}$ and $m_{N}$, and
took into account the LR gauge boson mixing.

We recalled the different signatures emerging as a function of the mass of the
RH neutrino $N$: from i) $\ell\ell jj$ in the KS process for $M_{W_{R}}\gtrsim
m_{N}\gtrsim 1\,\text{TeV}$, to ii) $\ell j_{N}$ in case the $N$ decay
products are merged in a single fat jet for few$\,\text{TeV}\gtrsim
m_{N}\gtrsim 100\,\text{GeV}$, to iii) the $\ell+\not{E}$ signature when $N$
decays outside the detector, for $m_{N}<300\,\text{GeV}$. All these channels
feature at least one prompt high $p_{T}$ lepton, which ensures triggering and
allows in reducing the expected SM background by many orders of magnitude. It
turns out that by requiring a minimal $p_{T}$ of the order of 1.5 TeV, the
background is dominated by `w+jets` and could be simulated to satisfactory
high statistics. We showed indeed that, as expected, the background and signal
live in different regions of kinematic observables, thus effectively leaving
just the signal at high energies. As a result, the reach is mainly limited by
the center of mass energy, luminosity and by quark PDFs.

We thus assessed the exclusion reach by adopting a unified binned likelihood
approach Nemevsek:2018bbt , which does not require sliding windows as a
function of model parameter choices. The results were presented in FIG. 10.
For the KS and merged regions, i) and ii), we estimated a reach for
$M_{W_{R}}$ as high as 35 TeV at 3 $\sigma$ C.L., for an integrated luminosity
of $3/$ab. This is similar to the reach expected for the simpler $W_{R}\to jj$
channel.

It is worth recalling here that within region ii), in the lightest $N$ regime
of $m_{N}\sim 100$–$300\,\text{GeV}$, the RH neutrino can decay at an
appreciable distance, giving rise to a _displaced jet_ Nemevsek:2018bbt . Its
study will be very interesting as soon as definite detector geometry and
sensitivities become available.

Still, this case overlaps with the even lower $m_{N}$ regime (case iii) where
$N$ decays outside the detector and appears as missing energy, thus matching
with the search for $W^{\prime}\to\ell\nu$. We estimated that the $3\sigma$
sensitivity covers a region extending up to $M_{W_{R}}\simeq 37,\text{TeV}$
and up to $m_{N}\simeq 300\,\text{GeV}$.

This region features an interesting connection with $0\nu\beta\beta$
contributions from the LRSM, especially in the light of current, planned and
envisioned experiments. FIG. 11 reports this interplay, showing that a
possible signal at forthcoming and future $0\nu\beta\beta$ probes, in the
absence of standard neutrino mass mechanism, would imply a $M_{W_{R}}$ below
$\sim 30\,\text{TeV}$ and $m_{N}$ below $\sim 100\,\text{GeV}$, which overlaps
precisely with the invisible $N$ decay at FCC here considered. Incidentally,
cosmology presents an interesting interplay, either as a constraint from BBN
or as an opportunity for having a warm DM candidate.

## Acknowledgments

MN is supported by the Slovenian Research Agency under the research core
funding No. P1-0035 and in part by the research grants J1-3013, N1-0253 and
J1-4389.

##  Appendix A Drell-Yan $W_{R}$ and $N$ production and decay rates

In this appendix we review in detail the production of $W_{R}$ and its decays
into the $\ell N$ final state (on shell and $2\to 2$).

###  A.1 Resonance production

The amplitude for the production of an on-shell resonance $q(p_{1})q(p_{2})\to
V(P)$ at parton level is

$\mathcal{A}=ig\,\overline{v}_{1}\gamma^{\mu}\left(v_{q}+a_{q}\gamma_{5}\right)u_{2}\varepsilon_{\mu}^{(\lambda)}.$
(10)

Averaging over initial spins, including the color factor and summing over
polarizations $\lambda$, we have

$\begin{split}\left|\mathcal{M}\right|^{2}&=\frac{N_{c}}{4}g^{2}\sum_{\lambda}\varepsilon_{\mu}^{(\lambda)}\varepsilon_{\nu}^{(\lambda)*}\text{tr}(\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta}\gamma^{\nu}\left(v_{q}+a_{q}\gamma_{5}\right))p_{1\alpha}p_{2\beta}\\\
&=\frac{N_{c}}{4}g^{2}\left(-g_{\mu\nu}+\frac{(p_{1}+p_{2})_{\mu}(p_{1}+p_{2})_{\nu}}{M^{2}}\right)4\left(p_{1}^{\mu}p_{2}^{\nu}+p_{1}^{\nu}p_{2}^{\mu}-g^{\mu\nu}p_{1}p_{2}\right)\left(v_{q}^{2}+a_{q}^{2}\right)\\\
&=N_{c}g^{2}\hat{s}\left(v_{q}^{2}+a_{q}^{2}\right),\end{split}$ (11)

where $\hat{s}=\left(p_{1}+p_{2}\right)^{2}=2p_{1}p_{2}$. The differential
cross-section is

$\text{d}\hat{\sigma}=(2\pi)^{4}\frac{\left|\mathcal{M}\right|^{2}}{2\,\hat{s}}\text{d}\varphi,$
(12)

such that integration over the final state momentum gives

$\displaystyle\hat{\sigma}$
$\displaystyle=\frac{\pi|\mathcal{M}|^{2}}{\hat{s}}\delta\left(\hat{s}-M^{2}\right)=4N_{c}\pi^{2}\alpha_{2}\left(v_{q}^{2}+a_{q}^{2}\right)\delta\left(\hat{s}-M^{2}\right).$
(13)

For the $W_{R}$ cross-section, we substitute $g\to gV_{ij}/\sqrt{2}$,
$v_{q}=a_{q}=1/2$ and obtain

$\displaystyle\hat{\sigma}_{q_{i}\overline{q_{j}}\to W_{R}}$
$\displaystyle=N_{c}\pi^{2}\alpha_{2}\left|V_{ij}\right|^{2}\delta\left(\hat{s}-M_{W_{R}}^{2}\right).$
(14)

Approximating the $\delta$ function with the Breit-Wigner resonance

$\pi\delta\left(\hat{s}-M^{2}\right)\simeq\frac{\Gamma
M}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma M\right)^{2}}\,,$ (15)

we get the parton level cross-section

$\hat{\sigma}_{q_{i}\overline{q_{j}}\to
W_{R}}=N_{c}\pi\alpha_{2}\left|V_{ij}\right|^{2}\frac{\Gamma_{W_{R}}M_{W_{R}}}{\left(\hat{s}-M_{W_{R}}^{2}\right)^{2}+\left(\Gamma_{W_{R}}M_{W_{R}}\right)^{2}}\,.$
(16)

Proton-proton. To obtain the $pp$ cross-section, the partonic $\hat{\sigma}$
is convoluted with the PDFs. There is an additional $1/N_{c}^{2}$
combinatorial factor for the color connection

$\frac{\text{d}^{2}\sigma_{pp\to
W_{R}^{+}}}{\text{d}x_{1}\text{d}x_{2}}=\frac{\pi^{2}\alpha_{2}}{N_{c}}\sum_{u,d}|V_{ud}|^{2}\left(f_{u}(x_{1})f_{\overline{d}}(x_{2})+1\leftrightarrow
2\right)\delta\left(\hat{s}-M_{W_{R}}^{2}\right).$ (17)

To integrate over $\hat{s}$, we change integration variables from $x_{1,2}$ to
the rapidity $y$ of $W_{R}$ and the partonic center of mass energy $\hat{s}$.
The proton mass is small and quarks are nearly massless, therefore

$\hat{s}=\left(p_{1}+p_{2}\right)^{2}=2p_{1}p_{2}=2x_{1}x_{2}P_{1}P_{2}=x_{1}x_{2}s\,.$
(18)

Moreover, proton beams are symmetric and energetic, such that the $W_{R}$
momentum is

$\displaystyle P_{W}^{0,3}$
$\displaystyle=x_{1}P_{1}^{0,3}+x_{2}P_{2}^{0,3}=\frac{\sqrt{s}}{2}\left(x_{1}\pm
x_{2}\right).$ (19)

This setup gives the $W_{R}$ rapidity

$\displaystyle
y\equiv\frac{1}{2}\ln\left(\frac{P_{W}^{0}-P_{W}^{3}}{P_{W}^{0}+P_{W}^{3}}\right)=\frac{1}{2}\ln\left(\frac{x_{2}}{x_{1}}\right).$
(20)

When $W_{R}$ is produced at rest, we have
$P_{W}^{3}=0=\sqrt{s}(x_{1}-x_{2})/2$, which implies $x_{1}=x_{2}$ and $y=0$.
The product of $x_{1}$ and $x_{2}$ is fixed by the on-shell condition for
$W_{R}$, which gives $P_{W}^{02}-P_{W}^{32}=M_{W_{R}}^{2}=x_{1}x_{2}s$. Now
the maximum value of $x_{2}^{\max}=1$ corresponds to
$x_{1}^{\min}=M_{W_{R}}^{2}/s$ and the maximal rapidity is

$\displaystyle y_{\max}=\ln\left(\frac{\sqrt{s}}{M_{W_{R}}}\right)\,.$ (21)

This also corresponds to the situation when $W_{R}$ is maximally boosted

$\displaystyle\gamma_{R}^{\max}=\frac{P_{W}^{0}}{M_{W_{R}}}=\frac{\sqrt{s}}{2}\left(x_{1}^{\min}+x_{2}^{\max}\right)=\frac{1}{2}\left(\frac{\sqrt{s}}{M_{W_{R}}}+1\right)\simeq\frac{\sqrt{s}}{2M_{W_{R}}}\,,$
(22)

where the last approximation is valid when $\sqrt{s}\gg M_{W_{R}}$. Because
quarks and protons are nearly massless, the $x_{1,2}$ have symmetric limits
and $y_{\min}=-y_{\max}$. We change variables using the Jacobian

$\text{d}x_{1}\text{d}x_{2}=\frac{\text{d}\hat{s}\,\text{d}y}{s}=x_{1}x_{2}\frac{\text{d}\hat{s}\,\text{d}y}{\hat{s}}\,,$
(23)

and finally end up with

$\frac{\text{d}\sigma}{\text{d}y}=\frac{\pi^{2}\alpha_{2}}{N_{c}M_{W_{R}}^{2}}\sum_{u,d}|V_{ud}|^{2}\left(x_{1}f_{u}(x_{1})x_{2}f_{\overline{d}}(x_{2})+1\leftrightarrow
2\right).$ (24)

Both partonic fractions $x_{1,2}$ are given by the collision energy and
$W_{R}$ rapidity $x_{1,2}=M_{W_{R}}e^{\pm y}/\sqrt{s}$. The distributions in
$y$ are symmetric at the LHC and extend to $\pm y_{\max}$.

###  A.2 Resonance Decays

The on-shell $W_{R}$ resonance can decay in different ways. The dominant decay
rates are

$\displaystyle\Gamma\left(W_{R}\to q_{R}\overline{q^{\prime}_{R}}\right)$
$\displaystyle=\frac{\alpha_{2}}{8}M_{W_{R}}\left(4+(2+x_{t})(1-x_{t})^{2}\right),$
(25) $\displaystyle\Gamma\left(W_{R}\to\ell_{R}N\right)$
$\displaystyle=\frac{\alpha_{2}}{24}\left|V_{\ell
N}\right|^{2}M_{W_{R}}\left(2+x_{N}\right)\left(1-x_{N}\right)^{2},$ (26)

where $x_{f}=1-m_{f}^{2}/M_{W_{R}}^{2}$, CKM is unitary and $V_{tb}\sim 1$.
Once we turn on the gauge boson mixing $\xi_{LR}$ in (1), the decay modes of
$WZ$ and $Wh$, as well as the SM $W$ decays to $\ell_{R}N$, open up. They
proceed with the following rates:

$\displaystyle\Gamma\left(W_{R}\to WZ\right)$
$\displaystyle=\Gamma\left(W_{R}\to
Wh\right)=\frac{\alpha_{2}}{48}s_{2\beta}^{2}M_{W_{R}}\,,$ (27)
$\displaystyle\Gamma\left(W\to\ell_{R}N\right)$
$\displaystyle=\xi_{LR}^{2}\frac{\alpha_{2}}{24}\left|V_{\ell
N}\right|^{2}M_{W}\left(2+x_{N}\right)\left(1-x_{N}\right)^{2}.$ (28)

At the same time the dominant rates in (25) and (26) get negligibly suppressed
by $1-\xi_{LR}^{2}$.

###  A.3 $\ell N$ fermion pair production via resonance

Instead of the $2\to 1$ resonance production above, we can consider the direct
$2\to 2$ scattering of $pp\to\ell N$ via the $W_{R}$ propagator. The
scattering amplitude in the unitary gauge is given by

$\begin{split}\mathcal{A}&=i\frac{g}{\sqrt{2}}V_{ij}\overline{u}_{1R}\gamma^{\mu}u_{2R}\frac{-ig_{\mu\nu}}{\hat{s}-M^{2}+i\Gamma
M}i\frac{g}{\sqrt{2}}V_{\ell N}\overline{u}_{3R}\gamma^{\nu}u_{4R}\\\
&=\frac{i}{2}\frac{g^{2}V_{ij}V_{\ell N}}{\hat{s}-M^{2}+i\Gamma
M}\left(\overline{u}_{1R}\gamma^{\mu}u_{2R}\right)\left(\overline{u}_{3R}\gamma_{\mu}u_{4R}\right),\end{split}$
(29)

where the momentum part of the $W_{R}$ propagator vanishes, because we take
quarks to be massless. Turning on the gauge boson mixing $\xi_{LR}$ does not
modify the production much, the $\overline{u}_{1R}\gamma^{\mu}u_{2R}$ term
goes into
$\cos(\xi_{LR})\overline{u}_{1R}\gamma^{\mu}u_{2R}+\sin(\xi_{LR})\overline{u}_{1L}\gamma^{\mu}u_{2L}$.
After squaring the amplitude and summing over the spins, we get
$\cos(\xi_{LR})^{2}+\sin(\xi_{LR})^{2}=1$, because the R and L terms sum into
the same expression, moreover the RL interference term vanishes in the limit
of zero quark masses. The spin averaged amplitude is

$\displaystyle\overline{\left|\mathcal{M}\right|}^{2}$
$\displaystyle=\frac{1}{16}N_{c}g^{4}\frac{\left|V_{ij}V_{\ell
N}\right|^{2}}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\text{tr}\left(\not{p}_{1}\gamma^{\mu}\not{p}_{2}\gamma^{\nu}P_{R}\right)\text{tr}\left(\not{p}_{3}\gamma_{\mu}\not{p}_{4}\gamma_{\nu}P_{R}\right),$
(30) $\displaystyle=4N_{c}\alpha_{2}^{2}\pi^{2}\frac{\left|V_{ij}V_{\ell
N}\right|^{2}}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\hat{t}\left(\hat{t}-m^{2}\right),$ (31)

where $\hat{t}=\left(p_{1}-p_{3}\right)^{2}=\left(p_{2}-p_{4}\right)^{2}$. The
partonic cross-section is then

$\displaystyle\hat{\sigma}_{q_{i}\overline{q_{j}}\to W_{R}^{*}\to\ell N}$
$\displaystyle=\frac{1}{16\pi\hat{s}^{2}}\int_{m^{2}-\hat{s}}^{0}\overline{\left|\mathcal{M}\right|}^{2}\,d\hat{t}$
(32)
$\displaystyle=\frac{1}{16\pi\hat{s}^{2}}4N_{c}\alpha_{2}^{2}\pi^{2}\frac{\left|V_{ij}V_{\ell
N}\right|^{2}}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\frac{1}{6}\left(\hat{s}-m^{2}\right)^{2}(2\hat{s}+m^{2})$ (33)
$\displaystyle=N_{c}\pi\alpha_{2}\left|V_{ij}\right|^{2}\frac{\sqrt{\hat{s}}}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\frac{\alpha_{2}}{24}\left|V_{\ell
N}\right|^{2}\sqrt{\hat{s}}\left(1-\frac{m^{2}}{\hat{s}}\right)^{2}\left(2+\frac{m^{2}}{\hat{s}}\right).$
(34)

In the narrow-width limit and expanding near the pole $\hat{s}\simeq M^{2}$,
the cross-section becomes

$\displaystyle\hat{\sigma}_{q_{i}\overline{q_{j}}\to\ell N}$
$\displaystyle=N_{c}\pi\alpha_{2}\left|V_{ij}\right|^{2}\frac{\Gamma
M}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\frac{\frac{\alpha_{2}}{24}\left|V_{\ell
N}\right|^{2}M\left(1-\frac{m^{2}}{M^{2}}\right)^{2}\left(2+\frac{m^{2}}{M^{2}}\right)}{\Gamma}$
(35) $\displaystyle\simeq
N_{c}\pi\alpha_{2}\left|V_{ij}\right|^{2}\pi\delta\left(\hat{s}-M^{2}\right)\frac{\Gamma_{W_{R}\to\ell
N}}{\Gamma}=\hat{\sigma}_{q_{i}\overline{q_{j}}\to
W_{R}}\text{Br}_{W_{R}\to\ell N}\,.$ (36)

Proton-proton. The partonic cross-section in (32) is convoluted with the PDFs.
The differential cross-section is

$\displaystyle\begin{split}\frac{\text{d}^{2}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\,\text{d}x_{2}}&=\frac{\pi\alpha_{2}^{2}}{24N_{c}}\frac{\hat{s}}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\left(1-\frac{m^{2}}{\hat{s}}\right)^{2}\left(2+\frac{m^{2}}{\hat{s}}\right)\times\\\
&\times\sum_{ud}\left|V_{ud}V_{\ell
N}\right|^{2}\left(f_{u}\left(x_{1}\right)f_{\overline{d}}\left(x_{2}\right)+1\leftrightarrow
2\right),\end{split}$ (37)

and we integrate over the $x_{1,2}$ to get the total cross-section

$\displaystyle\sigma_{pp\to\ell^{+}N}$
$\displaystyle=\int_{\frac{m^{2}}{s}}^{1}\text{d}x_{1}\int_{\frac{m^{2}}{x_{1}s}}^{1}\text{d}x_{2}\frac{\text{d}^{2}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\,\text{d}x_{2}}\,.$
(38)

The final state kinematic variables $p_{T}^{2}$ and $\eta_{\ell},\eta_{N}$
constrain the integration regions. Let

$\displaystyle p_{1}$
$\displaystyle=x_{1}\frac{\sqrt{s}}{2}\left(1,0,0,1\right),$ $\displaystyle
p_{3}$
$\displaystyle=\left(m_{T}\cosh\eta_{N},0,p_{T},m_{T}\sinh\eta_{N}\right),$
(39) $\displaystyle p_{2}$
$\displaystyle=x_{2}\frac{\sqrt{s}}{2}\left(1,0,0,-1\right),$ $\displaystyle
p_{4}$
$\displaystyle=\left(p_{T}\cosh\eta_{\ell},0,-p_{T},-p_{T}\sinh\eta_{\ell}\right),$
(40)

where the transverse mass is defined by $m_{T}^{2}=m^{2}+p_{T}^{2}$. At high
momentum transfers considered here, the proton and charged lepton masses are
negligible and $p_{1}^{2}=p_{2}^{2}=p_{4}^{2}=0$. However, we treat the heavy
Majorana neutrino as massive with a mass $p_{3}^{2}=m^{2}$. From these, we get
the $x_{1,2}$ and the Mandelstam invariants, $\hat{t}=(p_{1}-p_{3})^{2}$ and
$\hat{s}=(p_{1}+p_{2})^{2}$, to be

$\displaystyle x_{1,2}$
$\displaystyle=\frac{1}{\sqrt{s}}\left(m_{T}e^{\pm\eta_{N}}+p_{T}e^{\mp\eta_{\ell}}\right),$
$\displaystyle\hat{t}$
$\displaystyle=-p_{T}\left(p_{T}+m_{T}e^{-\eta_{N\ell}}\right),$
$\displaystyle\hat{s}$ $\displaystyle=x_{1}x_{2}s\,,$ (41)

where $\eta_{N\ell}=\eta_{N}+\eta_{\ell}$. The $p_{T}$ and rapidities are
functions of $x_{1,2}$ ($s$ if fixed) and $\hat{t}$

$\displaystyle p_{T}^{2}$
$\displaystyle=\frac{-\hat{t}\left(\hat{t}+\hat{s}-m^{2}\right)}{\hat{s}}$
$\displaystyle e^{2\eta_{N}}$
$\displaystyle=-\frac{x_{1}}{x_{2}}\left(1+\frac{\hat{s}}{\hat{t}}\right)$
$\displaystyle e^{2\eta_{\ell}}$
$\displaystyle=-\frac{x_{2}}{x_{1}}\left(1+\frac{\hat{s}}{\hat{t}-m^{2}}\right)$
(42) $\displaystyle=\frac{-\hat{t}\left(\hat{t}+\tau s-m^{2}\right)}{\tau
s}\,,$ $\displaystyle=-\frac{1}{\rho}\left(1+\frac{\tau s}{\hat{t}}\right),$
$\displaystyle=-\rho\left(1+\frac{\tau s}{\hat{t}-m^{2}}\right),$ (43)

where we introduced

$\displaystyle\tau$ $\displaystyle=x_{1}x_{2}\,,$ $\displaystyle\rho$
$\displaystyle=\frac{x_{2}}{x_{1}}\,.$ (44)

The $\tau$ variable is useful because it corresponds to the invariant mass of
the $\ell N$ pair, namely $m_{\text{inv}}^{2}(\ell
N)=(p_{3}+p_{4})^{2}=\hat{s}=x_{1}x_{2}s=\tau s$. These variables simplify the
imposition of cuts and efficiencies. From the matrix element in (30) and the
definition of the cross-section in (32), we have

$\displaystyle\frac{\text{d}^{3}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\,\text{d}x_{2}\,\text{d}\hat{t}}$
$\displaystyle=\frac{\pi\alpha_{2}^{2}}{12\,\hat{s}^{2}}\frac{\hat{t}\left(\hat{t}-m^{2}\right)}{\left(\hat{s}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\sum_{ud}\left|V_{ud}V_{\ell
N}\right|^{2}\left(f_{u}(x_{1})f_{\overline{d}}(x_{2})+1\leftrightarrow
2\right).$ (45)

Invariant mass. The $m_{\text{inv}}\in[m,\sqrt{s}]$. Taking into account that
$m_{\text{inv}}^{2}=\hat{s}=x_{1}x_{2}s=\tau s$, we get that

$\displaystyle\frac{\text{d}\sigma_{pp\to\ell^{+}N}}{\text{d}m_{\text{inv}}}$
$\displaystyle=\frac{\text{d}\sigma_{pp\to\ell^{+}N}}{\text{d}m_{\text{inv}}^{2}}\frac{\text{d}m_{\text{inv}}^{2}}{\text{d}m_{\text{inv}}}=\frac{1}{s}\frac{\text{d}\sigma_{pp\to\ell^{+}N}}{\text{d}\tau}2m_{\text{inv}}=\frac{2m_{\text{inv}}}{s}\int_{m_{\text{inv}}^{2}/s}^{1}\frac{\text{d}x_{1}}{x_{1}}\frac{\text{d}^{2}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\text{d}x_{2}}$
(46)
$\displaystyle\begin{split}&=\frac{\pi\alpha_{2}^{2}}{12N_{c}s}\frac{m_{\text{inv}}^{3}}{\left(m_{\text{inv}}^{2}-M^{2}\right)^{2}+\left(\Gamma
M\right)^{2}}\left(1-\frac{m^{2}}{m_{\text{inv}}^{2}}\right)^{2}\left(2+\frac{m^{2}}{m_{\text{inv}}^{2}}\right)\times\\\
&\times\sum_{ud}\left|V_{ud}V_{\ell
N}\right|^{2}\int_{m_{\text{inv}}^{2}/s}^{1}\frac{\text{d}x_{1}}{x_{1}}\left(f_{u}\left(x_{1}\right)f_{\overline{d}}\left(x_{2}\right)+1\leftrightarrow
2\right),\end{split}$ (47)

where $x_{2}=m_{\text{inv}}^{2}/(x_{1}s)$ and the Jacobian from
$(x_{1},x_{2})\to(x_{1},\tau)$ is equal to $1/x_{1}$.

Transverse momentum. Along the same lines, the $p_{T}$ distribution is
obtained by the chain rule

$\displaystyle\frac{\text{d}\sigma_{pp\to\ell^{+}N}}{\text{d}p_{T}}$
$\displaystyle=\int_{x_{1,2}}\frac{\text{d}^{3}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\,\text{d}x_{2}\,\text{d}\hat{t}}\frac{\text{d}\hat{t}}{\text{d}p_{T}}\,,$
(48) $\displaystyle\frac{\text{d}\hat{t}}{\text{d}p_{T}}$
$\displaystyle=\frac{2p_{T}\hat{s}}{\sqrt{(\hat{s}-m^{2})^{2}-4p_{T}^{2}\hat{s}}}\,,$
(49) $\displaystyle\hat{t}\left(\hat{t}-m^{2}\right)$
$\displaystyle=\frac{\hat{s}}{2}\left(\hat{s}-m^{2}-2p_{T}^{2}+\sqrt{(\hat{s}-m^{2})^{2}-4p_{T}^{2}\hat{s}}\right).$
(50)

This gives us the distribution over $p_{T}$, shown in Fig. 3. For
$\hat{t}\in\mathbb{R}$, the argument of the square root in (50) needs to be
positive, which leads to an upper bound on $p_{T}$

$\displaystyle p_{T}\in\left[0,\frac{s-m^{2}}{2\sqrt{s}}\right].$ (51)

Furthermore, at a fixed value of $p_{T}$, the lower bound for
$\tau=x_{1}x_{2}$ is given by

$\displaystyle\tau_{0}=\frac{m^{2}+2p_{T}^{2}+2\sqrt{p_{T}^{2}(m^{2}+p_{T}^{2})}}{s}\xrightarrow{p_{T}\to
0}\frac{m^{2}}{s}\,,$ (52)

and the integration limits in (48) are given by

$\displaystyle\int_{x_{1,2}}=\int_{\tau_{0}}^{1}\text{d}x_{1}\int_{\tau_{0}/x_{1}}^{1}\text{d}x_{2}\,.$
(53)

Invariant mass vs $p_{T}$. Likewise, we can get the double differential
distribution over $p_{T}$ and $m_{\text{inv}}$ by combining the two chain
rules and integrating over $x_{1}$

$\displaystyle\frac{\text{d}^{2}\sigma_{pp\to\ell^{+}N}}{\text{d}p_{T}\,\text{d}m_{\text{inv}}}$
$\displaystyle=\frac{2m_{\text{inv}}}{s}\int_{m_{\text{inv}}^{2}/s}^{1}\frac{\text{d}x_{1}}{x_{1}}\,\frac{\text{d}^{3}\sigma_{pp\to\ell^{+}N}}{\text{d}x_{1}\,\text{d}x_{2}\,\text{d}\hat{t}}\,\frac{\text{d}\hat{t}}{\text{d}p_{T}}\,,$
(54)

using (45), (49) and (50), where again $\hat{s}=m_{\text{inv}}^{2}$ and
$x_{2}=m_{\text{inv}}^{2}/(x_{1}s)$. From (49) we see that

$\displaystyle p_{T}$
$\displaystyle\in\left[0,\frac{m_{\text{inv}}^{2}-m^{2}}{2m_{\text{inv}}}\right],\qquad\text{
for a fixed }\quad m_{\text{inv}}\in\left[m,s\right].$ (55)

The contours of the double differential cross-section are interesting in that
they are dominated by a narrow diagonal line relative to off-shell $W_{R}$
production, and a some broader regions with lower $p_{T}$ but fixed
$m_{\text{inv}}\simeq M_{W_{R}}$, corresponding to on-shell $W_{R}$.

No cuts

Without cuts, the lower bound on $\tau$ is simply $\tau_{0}=m^{2}/s$, as
needed to produce a massive $N$. However, the integration limits for $\rho$
are split into two regions

$\tau\in[\tau_{0},\rho],\text{ when }\rho\in[\tau_{0},1]\text{ and
}\tau\in[\tau_{0},1/\rho]\text{ for }\rho\in[1,1/\tau_{0}]\,.$ (56)

Meanwhile, $\hat{t}\in[m^{2}-\tau s,0]$.

$p_{T}$ cut

$p_{T}>p_{Tc}$ can be implemented within the integration limits. Note that
$\eta_{\ell N}$ does not depend on $p_{T}$. From the $p_{T}$ equation in (43),
we have

$\hat{t}_{\pm}(\tau)=\frac{s}{2}\left(\tau_{0}-\tau\right)\left(1\pm\sqrt{1-\frac{p_{Tc}^{2}}{s}\frac{4\tau}{(\tau_{0}-\tau)^{2}}}\right),$
(57)

and from the positivity of the square root, a $\rho$-independent constant
lower bound

$\tau_{-}=\tau_{0}+\frac{2p_{Tc}^{2}}{s}\left(1+\sqrt{1+\frac{m^{2}}{p_{Tc}^{2}}}\right)\xrightarrow{p_{Tc}\to
0}\tau_{0}\,,$ (58)

appears. The $\tau-\rho$ integration plane is $\tau\in[\tau_{-},\rho]$, when
$\rho\in[\tau_{-},1]$ and $\tau\in[\tau_{-},1/\rho]$ for
$\rho\in[1,1/\tau_{-}]$.

$\eta_{\ell}$ cut

restriction $\left|\eta_{\ell}\right|>\eta_{\ell c}$ makes the $\tau_{-}$
limit $\rho$ dependent. Notice that $\hat{t}_{\min}=0$, so setting $\hat{t}=0$

$\tau_{-}(\rho)=\tau_{0}\left(1+\frac{e^{-2\eta_{\ell
c}}}{\rho}\right)\xrightarrow{\eta_{\ell c}\to\infty}\tau_{0},$ (59)

while the $\rho$ interval comes from $\tau_{-}(\rho_{-})=\rho_{-}$, and
$\tau_{-}(1/\rho_{+})=1/\rho_{+}$, such that

$\rho\in[\rho_{-},\rho_{+}]=\left[\frac{\tau_{0}}{2}\left(1+\sqrt{1+\frac{4e^{-2\eta_{\ell
c}}}{\tau_{0}}}\right),\frac{1}{\tau_{0}}-e^{-2\eta_{\ell c}}\right].$ (60)

Finally, the bound on $\hat{t}$ coming from (43) is

$\hat{t}(\tau)=s\left(\tau_{0}-\frac{\tau}{1+e^{-2\eta_{\ell
c}}/\rho}\right),\quad-\hat{t}_{\max}=-\hat{t}(\rho=1)=s\left(\tau_{0}-\frac{1}{1+e^{-2\eta_{\ell
c}}}\right).$ (61)

$p_{T}$ and $\eta_{\ell}$

With both cuts acting simultaneously, the integration limits become more
complex. The upper bound on $\tau(\rho)$ remains the same, however the lower
bound $\tau_{-}$ depends on both, $p_{T}$ and $\eta_{\ell}$ cuts. More
importantly, the $\rho$ interval changes as well as the limits on $\hat{t}$.

Let us start with the lower bound on $\tau$. Solving the quadratic equation
for $\hat{t}$ in (43) and plugging into $\eta_{\ell}$, we have

$\tau_{-}^{\pm}(\rho)=\left(m^{2}+p_{Tc}^{2}\right)\left(1+\frac{e^{\pm
2\eta_{\ell c}}}{\rho}\right)+p_{Tc}^{2}\left(1+e^{\mp 2\eta_{\ell
c}}\rho\right).$ (62)

The $+(-)$ applies to regions of $\rho$ above (below) 1, where the upper bound
is $\rho(1/\rho)$, such that

$\displaystyle\rho^{b}_{+}=-\frac{e^{\eta_{\ell c}}}{2p_{Tc}^{2}}$
$\displaystyle\left(e^{\eta_{\ell
c}}\left(m^{2}+2p_{Tc}^{2}\right)+\sqrt{e^{2\eta_{\ell
c}}m^{4}+4p_{Tc}^{2}s}\right),$ (63)
$\displaystyle\rho^{b}_{-}=-\frac{e^{-\eta_{\ell
c}}}{2(p_{Tc}^{2}e^{2\eta_{\ell c}}-s)}$ $\displaystyle\left(e^{\eta_{\ell
c}}\left(m^{2}+2p_{Tc}^{2}\right)+\sqrt{e^{2\eta_{\ell
c}}m^{4}+4\left(p_{Tc}^{2}+m^{2}\right)s}\right).$ (64)

The lower bound in (62) should not go below the $\rho$ independent one in
(58), which happens at

$\displaystyle\rho^{c}_{\pm}=e^{\pm\eta_{\ell
c}}\sqrt{1+\frac{m^{2}}{p_{Tc}^{2}}}.$ (65)

Notice that above a limiting $p_{T}$ cut

$p_{Tc}^{2\text{ lim}}=e^{-2\eta_{\ell c}}s-m^{2},$ (66)

the $\eta_{\ell}$ cut becomes ineffective (we are back to the $p_{T}$ cut case
above) and (66) implies an upper limit on $\eta_{\ell
c}^{\text{lim}}=-\log(m/\sqrt{s})$.

$N$ as missing energy. Let us compute the number of events $n_{\ell\not{E}}$,
when $N$ decays outside of the detector of size $l_{0}\sim 5\text{ m}$. Events
are distributed by an exponential distribution

$\displaystyle\frac{\text{d}n}{\text{d}l}$
$\displaystyle=n_{0}\frac{\exp(-l/L)}{L}\,,$ $\displaystyle L$
$\displaystyle=\gamma_{N}\beta_{N}\tau_{N}=\frac{p_{T}}{m\,\Gamma_{N}}\sqrt{1+\left(1+\frac{m^{2}}{p_{T}^{2}}\right)\sinh\eta_{N}^{2}}\,,$
(67)

and the total number of events is obtained by integrating from $\ell_{0}$ to
$\infty$

$\displaystyle n_{\ell\not{E}}$
$\displaystyle=\int_{l_{0}}^{\infty}\frac{\text{d}n}{\text{d}l}\text{d}l=\mathcal{L}\iiint\frac{\text{d}\rho\text{d}\tau\text{d}\hat{t}}{2\rho}\varepsilon_{\ell}(p_{T},\eta_{\ell})\left(\frac{\text{d}^{3}\sigma_{pp\to\ell
N}}{\text{d}\tau\,\text{d}\rho\,\text{d}\hat{t}}\right)e^{-l_{0}/L}\,,$ (68)

where $\mathcal{L}$ is the total luminosity and $\varepsilon_{\ell}$ is the
charged lepton selection efficiency. Efficiencies may also include more
stringent $p_{T}$ and $\eta_{\ell}$ cuts.

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